Martingales and Profile of Binary Search Trees

We are interested in the asymptotic analysis of the binary search tree (BST) under the random permutation model. Via an embedding in a continuous time model, we get new results, in particular the asymptotic behavior of the profile.


ST) is a stru
ture used in computer science to store totally ordered data.At time 0 the LBST is reduced to a leaf without label.Each unit of time, a ew item is inserted in a leaf of the tree.This leaf is then replaced by an internal node with two leaves.We are interested in the sequence of underlying unlabeled trees (T n ) n induced by this construction.We call this sequence the binary search tree process, or BST process.

• The Yule tree process (T T t ) t is a continuous time (unlabeled) binary tree process in which each leaf behaves independently from the other ones (at time 0, the tree T T 0 is reduced t a leaf).After an (random) exponential time, a leaf has two children.Due to the lack of memory of the exponential distribution, each leaf is equally likely the first one to produce children.

Under a suitably chosen random model of data (the random permutation model), the two models of trees are deeply related.In the Yule tree process, let τ n be the random time when the n+1th le f appears.Under the random permutation model the link between the two models is the following one: the process (T T τn ) n has the same law as (T n ) n .This allows the construction of the BST process and the Yule tree process on the same probability space on which (T n ) n = (T T τn ) n .This embedding of the BST process into a continuous time model allows to use independence properties between subtrees in the Yule tree process (it is a kind of Poissonization).Many functionals of the BST can then be derived using known results on the Yule tree.An interesting quantity is the profile of T n which is the sequence (U k (n)) k≥0 where U k (n) is the number of leaves of T n at level k.Here, in (21), the martingale family (M n (z)) n -the Jabbour's martingale -which encodes the profile of (T n ) n is shown to be strongly related to the martingale family (M t (z)) t that encodes the profile of (T T t ) t .


The models 2.1 Binary search trees

For a convenient definition of trees we are going to work with, let us first define
U = {∅} ∪ n≥1 {0, 1} n
the set of finite words on the alphabet {0, 1} (with for the empty word).For u and v in U, denote by uv the concatenation of the word u with the word v (by convention we set, for any u ∈ U, ∅u = u).If v = ∅, we say that uv is a descendant of u and u is an ancestor of uv.Moreover u0 (resp.u1) is called left (resp.right) child of u.

A complete binary tree T is a finite subset of U such that
   ∅ ∈ T if uv ∈ T then u ∈ T , u1 ∈ T ⇔ u0 ∈ T .
The elements of T are called nodes, and ∅ is called the root ; |u|, the number of letters in u, is the depth of u (with |∅| = 0).Write BinTree for the set of complete binary trees.

A tree T ∈ BinTree can be described by giving the set ∂T of its leaves, that is, the nodes that are in T but with no descend nts in T .The nodes of T \∂T are called internal nodes.

We now introduce labeled binary search trees (LBST), that are widely used to store totally ordered data (the monograph of Mahmoud [31] gives an overview of the state of the art).

Let A be a totally ordered set of elements named keys and for n ≥ 1, let (x 1 , ..., x n ) be picked up without replacement from A. The LBST built from these data is the complete binary tree in which each internal node is associated with a key belonging to (x 1 , ..., x n ) in the following way: the first key x 1 is assigned to the root.The next key x 2 is assigned to the left child of the root if it is smaller than x 1 , or it is assigned to the right child of the root if it is larger than x 1 .We proceed further inserting key by key recursively.After the n first insertion, one has a labeled binary tree in which n nodes own a label: hese nodes are considered as internal nodes.One adds n + 1 (unlabeled) leaves to this structure in order to get a labeled complete binary tree with n internal nodes.Figure 1: BST built with the sequence of data 0.5, 0.8, 0.9, 0.3, 0.4 (empty squares are leaves).

To study the shape of these trees for large n, it is classical to introduce a ra dom model.One u

ally assumes that the s
ccessively inserted data (x i ) i≥1 are i.i.d.random variables with a continuous dis nder this model, let us call the LBST L n , but that has no label.We set
T (F ) n , n ≥ 0 := UNDER(L (F ) n ), n ≥ 0 ; by construction T (F ) n
is a complete binary tree.For every n ≥ 1, the string x 1 , .., x n induces a.s. a permutation σ n such that x σn(1) < x σn(2) < • • • < x σn(n) .Since the x i are exchangeable, σ n is uniformly distribute on the set S n of permutations of {1, .., n}.Since this cl e of simplicity, that F is the uniform distribution on [0, 1], and we write from now
L n instead of L (F ) n and T n instead of T (F )
n .This is the so-called random permutation model.Again by exchangeability, σ n is independent of the vector (x σn(1) , x σn(2) , . . ., x σn(n) ) and we have
P x n+1 ∈ (x σn(j) , x σn(j+1) ) | σ n = P x n+1 ∈ (x σn(j) , x σn(j+1 ) = P (σ n+1 (j + 1) = n + 1) = (n + 1) −1
for every j ∈ {0, 1, .., n}, where x σn(0) := 0 and x σn(n+1) := 1.This relation ensures the consistency of the sequence (σ n ) n .

On can also express this property with the help of the sequential ranks of the permutation: the random variables R k = k j=1 1I x j ≤x k , k ≥ 1 are independent and R k is uniform on {1, . . ., k} (see for instance Mahmoud [31], section 2.3), so that
P (R n+1 = j + 1 | R 1 , .., R n ) = (n + 1) −1 .
In terms of binary search tree, this means that the insertion of the n + 1st key in the tree with n internal nodes is uniform among its n + 1 leaves.In other words, in the random permutation model, the sequence (T n ) n≥0 is a Markov chain on BinTree defined by T 0 = {∅} and
T n+1 = T n ∪ {D n 0, D n 1} , P (D n = u | T n ) = (n + 1) −1 , u ∈ ∂T n ;(1)
the leaf D n of T n is the random node where the n + 1-st key is inserted, its level is d n .

The difference of the rule evolutions of L n (that depends deeply on the values x 1 , . . ., x n already inserted) and T n (that depends of nothing) is similar to Markov chain in random environment (L n is the quenched Markov chain and T n the annealed one).

This Markov chain model is a particular case (α = 1) of the diffusion-limited aggregation (DLA) on a binary tree, where a constant α is given and the growing of the tree is random with probability of insertion at a leaf u proportional to |u| −α (Aldous-Shields [1], Barlow-Pemantle-Perkins [6]).

Here are few known results about the evolution of BST.First, the saturation level h n and the height H n ,
h n = min{|u| : u ∈ ∂T n } Devroye [17] )
a.s. lim n→∞ h n log n = c ′ = 0.3733... lim n→∞ H n log n = c = 4.31107... ;(3)
the constants c ′ and c are the two solutions of the equation η 2 (x) = 1 where
η λ (x) := x log x λ − x + λ, x ≥ 0 ,(4)
is the Cramer transform of the Poisson distribution of parameter λ.Function η 2 reaches its minimum at x = 2.It corresponds to the rate of propagation of the depth of insertion: dn 2 log n P −→ 1.More precise asymptotics for H n can be found in [19], [36], [37], [28].

Detailed information on T n is provided by the whole profile
U k (n) := #{u ∈ ∂T n , |u| = k} , k ≥ 1 ,(5)
that counts the number of leaves of T n at each level.Notice that U k (n) = 0 for k > H n and for k < h n .To get asymptotic results, it is rather natural to encode the profile by the so-called polynomial level k U k (n)z k , whose degree is H n .Jabbour [15,25] proved a remarkable martingale property for these random polynomials.More precisely, for z / ∈
1 2 Z Z − = {0, −1/2, −1, −3/2, • • • } and n ≥ 0, let M n (z) := 1 C n (z) k≥0 U k (n)z k = 1 C n (z) u∈∂Tn z |u| ,(6)
where C 0 (z) = 1 and for n ≥ 1,
C n (z) := n−1 k=0 k + 2z k + 1 = (−1) n −2z n ,(7)
and let F (n be the σ-field generated by all the events {u ∈ T j } j≤n,u∈U .Then (M n (z), F (n) ) n is a martingale to which, for the sake of simplicity, we refer from now as the BST martingale.If z > 0, this positive martingale is a.s.convergent; the limit M ∞ (z) is positive a.s.if z ∈ d M ∞ (z) = 0 for z / ∈ [z − c , z + c ] (Jabbour [25]).This martingale is also the main tool o prove that, properly rescaled around 2 log n, the profile has a Gaussian limiting shape (see Theorem 1 in [15] ).


Fragmentation, Yule tree process and embedding

The idea of embedding discrete models (such as urn models) in continuous time branching processes goes back at least to Athreya-Karlin [4].It is described in Athreya and Ney ( [5], section 9) and it has been recently revisited by Janson [26].For the BST, various embeddings are mentioned in Devroye [17], in particular those due to Pittel [35], and Biggins [12,13].Here, we work with a vari nt of the Yule process, taking into account the tree (or "genealogical") structure.

First, let us define a fragmentation process (F (t)) t≥0 of the interval (0, 1) = u 1 u 2 ...u k ∈ U, set I u the interval
I u = k j=1 u j 2 −j , 2 −k + k j=1 u j 2 −j .
Hence, each element u of U encodes a subinterval I u of (0, 1) with dyadic extremities.

We set F (0) = I ∅ = (0, 1).An exponential τ 1 ∼ Exp(1) random variable is associated with F (0).At time τ 1 , the process F. jumps, the interval (0, 1) s ) = ((0, 1/2), (1/2, 1)) = (I 0 , I 1 ).After each jump time τ , the fragments of F (τ ) behave independently of each other.Each fragment I u splits after a Exp(1)-distributed random time into two fragments: I u0 and I u1 .Owing to the lack of memory of the exponential distribution, when n fragments are present, each of them will split first equally likely.

We define now the Yule tree process as an encoding of th s I u0 and I u1 issued from I u a ragment and I u1 the right one; like this, we obtain a binary tree structure (see Fig. 2).An interval with length 2 −k corresponds to a leaf at depth k in the corresponding tree structure; the size of fragment I u is 2 −|u| .More formally, we define the tree T T andom process (T T t ) t≥0 .Both processes (T T t ) t≥0 and (F (t)) t≥0 are pure jump Markov processes.Each process (T T t ) t≥0 and (F (t)) t≥0 can be viewed as an encoding of the other one, using (9) and:
F (t) =

I u , u ∈ ∂T T t }.
The counting process (N t )
t≥0 that gives the number of leaves in T T t ,
N t := #∂T T t ,(10)
is the classical Yule (or binary fission) process (Athreya-Ney [5]).Let 0 = τ 0 < τ 1 < τ 2 < ... be the successive jump times of T T. (or of (F (.)),
τ n = inf{t : N t = n + 1} .(11)
The following proposition allows us to build the Yule tree process and the BST on the same probability space.This observation was also made in Aldous-Shields [1] section 1, (see also Kingman [27] p.237 and Tavaré [40] p.164 in other contexts).


Lemma 2.1

a) The jump time intervals (τ n − τ n−1 ) n are independent and satisfy:
τ n − τ n−1 ∼ Exp(n) for any n ≥ 1, (12)
where Exp(λ) is the e processes (τ n ) n≥1 and T T τn n≥1 are independent.

c) The processes T T τn n≥0 and T n n≥0 have the same distribution.

Proof: (a) is a consequence of the fact that the minimum of n independent random variables Exp(1)-distributed is Exp(n)-distributed.(b) comes from the independence of jump chain and jump times.Since the initial states and evolution rules of the two Markov chains T T τn and T n are the same ones, (c) holds true.

Convention: (A unique probability space) From now, we consider that the fragmentation process, the Yule tree process and the BST process are built on the same probability space.Particularly, on this space, we have T τn n≥0 = T n n≥0 .

We say that the BST process is embedded in the Yule tree process.We define the filtration (F t ) t≥0 by F t = σ(F (s), s ≤ t).On the unique probability space, the sigma algebra
F (n) is equal to σ(F (τ 1 ), . . . , F (τ n )).
If we consider the measure valued process (ρ t ) t≥0 defined by
ρ t = u∈∂T Tt δ − log 2 |Iu| = u∈∂T Tt δ |u| ,(14)
we obtain a continuous time branching random walk.The set of positions is
IN 0 = {0, 1, 2, • • • }.
Each individual has an Exp(1) distribute t his death, he disappears and is replaced by two children, whose positions are both their parent's position shifted by 1.The set of individuals alive at time t is ∂T T t and the position of individual u is simply |u|.This is a particular case of the follo cal measure of the logarithm of the size of fragments in homogeneous fragme dislocation measures is a branching random walk (this idea goes back o Aldous and Shields [1] Section 7f and 7g).


Martingales and connection

The class artingales associated with the Yule process, parameterized by θ in IR (sometimes in ere given by
m(t, θ) := u∈∂T Tt exp(θ|u| − tL(θ)),
where
L(θ) = 2e θ − 1(15)
(see [41], [29], and [9] for the fragmentation).For easier use, we set z = e θ and then consider the family of (F t , t ≥ 0)-martingales
M (t, z) := m(t, log z) = u∈∂T

z |u| e t
1−2z) .(16)
In particular M (t, 1/2) = 1 and M (t, 1) = e −t N t .The emb e family of BST martingales (M n , F (n) ) n to the family of Yule martingales (M (t, z), F t ) t .If we observe the marti gale (M (., z) at the stopping times (τ n ) n , we can "extract" (Pr position 2.2 below) the space component M n (z) and a time component
n (z) := e τn(1−2z) C n (z) .(17)
Notice that n (z) n is F τn -adapted.A classical result (see Athreya-Ney [5] or Devroye [17] 5.4) says that, a.s., e −t N t converges when t → +∞, and
ξ := lim t→∞ e −t N t ∼ Exp(1) . (18)
Since lim n τ n = ∞ a.s (see Lemma 2.1 a) ) we get from ( 11) and ( 18), a.s.lim
n ne −τn = ξ . (19) Proposition 2.2 (martingale connection) Let us assume z ∈ C  \ 1 2 Z Z − . 1)
The family n (z) n≥0 is a martingale with mean 1, and a.s.
lim n n (z) = ξ 2z−1 Γ( z) . (20)
Moreover, if ℜz, the real part of z, is positive, the convergence is in L 1 .

2) (M n (z)) n≥0 are independent and
M (τ n , z) = n (z)M n (z) .(21)
Proof: 1) The martingale propert comes from Lemma 2.1 a).The Stirling formula gives the very use elds (20) owing to (19).

2) The second claim comes from ( 13) and ( 16), the independence comes from Lemma 2.1 b).

Proposition 2.2 allows us to transfer known results about the Yule martingales to BST martingales, thus giving very simple proofs of known results about the BST martingale and also getting much more.In particular, in Theorem 3.3 2), we give the answer to the question asked in [25], about critical values of z, with a straightforward argument.


Limiting proportions of nodes

Let us study some meaningful random variables arising as a.s limits and playing an important role in the results of Section 3.These variables describe the evolution of relative

get the represent
tion
a.s. lim t→∞ n (u) t N t = v<u U (v) ,(28)
where the random variables (U (v) ) v∈U satisfy the claim.This is of cou e BST

It is straightforward to see that, by embedding, the property of the above subsection holds true for limiting proportions of nod the BST).Let us now sketch the argument for LBST.

Assume x 1 fixed.Consider the tree e n data x 2 , . . ., x n+1 .Let K(n) := #{i, i ∈ 2, n + 1 , x e the x i are i.i.d., U [0, 1], the conditional distribution of K(n) on x 1 , is a binomial B(n, x 1 ).Hence, by the strong law of large numbers,
K(n) n a.s. − − → btree t 1 rooted in u = 1 a is build with the ones that are larger than x 1 ).In particular, the label x t 0 value a niform on [0, x 1 ], therefore it has the following representation: x t 0 = x 1 U where U is uniform on [0, 1] and does not depend on the value x 1 .Hence, the asymptotic proportio it is x 1 (1 − U ) in t 01 (what happens in the subtree t 1 is totally independent).This iterative construction of the LBST explains why it enjoys the same property as (28) in the Yule process, and so does the sequence of underlying BST T n .This is a strong, which means a.s., version of the analogy between BST and bra asymptotic behaviors of the Yule and BST martingales.The martingale connection (Proposition 2.2) allows to express the links between the limits.


Additive martingales

Theorem 3.1 gives an answer to a natural question asked in [15] about the domain in the complex plane where the BST martingale is L 1 −convergent nd uniformly convergent.Theorem 3.4 gives the optimal L 1 domain on R.
Theorem 3.1 For 1 < q ∞}. Then V q = {z : f (z, q) > 0} with f (z, q) := 1 + q(2ℜz − 1) − 2|z| q . (29)
If we denote V := ∪ 1<q<2 V q , we have : a) A

ves in the Yule tree i
s of [41], the exponential rate of growing is ruled by the function
x → L ⋆ (x) := sup θ θx − L(θ) =

η = 0 . (58) Write M (t, z) = e t(1−2z) k
−2tz(1−e iη ) e −ikη dη
and, owing to Lemma 4.3
2πρ t (k)e t(1−2z) z k √ t niform in k and in z in any compact subset of (z − c , z + c ).Now, from the Cauchy formula we get that
π −π e −2zt(1−e iη ) e −ikη dη = 2πe −2zt (2zt) k k! , yielding(56)
, which ends the proof.


Tagged branches and bi tting of the Yule and BST processes.This pro imes.On hes.T that the whole tree owns a different behavior.This method is usual and fruitful in modern developments on branching processes, lting method in the setting of BST provides new tools to study some ch ability to pass from a tilted model to the non-t

ted model: they app
ar as Radon-Nikodym derivatives.The parameter z, present in the martingales (M (t, z)) t≥0 and (M n (z)) n≥0 , allows to tune the growing of the special ray, changing in a visible way the shape of

he (Yule o
BST) tree.


Tilted fragmentation and biased Yule tree

First at all, let us enlarge the probability space of th val fr t and V be a U ([0, 1]) r.v.independent k ∈ N}) = 0, we may def ) and V ∈ I S(t) .In other words, S(t) is the element of U encoding the fragment containing V , its depth is s(t) := |S(t)|, the length of I S(t) is 2 −s(t) and
P(S(t) = u | F t ) = 2 −|u| , u ∈ ∂T T t (59)
(

is equivalent to choose a fragme
t at random with probability equal to its length, it is the classical size-biasing setting).Now we build the process ( T T t ) t≥0 of marked binary Yule trees associated with the pair (F (.), S(.)).The only change with Section 2.2 is the role played by the random variable V (missing in Section 2.2).During the construction of the Yule tree, at any given time t, each leaf in T T t corresponds to an interval in the fragmentation F (t).For every t we mark the leaf S(t) of T T t that corresponds to the interval I S(t) that contains V .We obtain a marked tree called ( T T t ) t≥0 .Thus, the set of nodes marked during [0, t] are the prefixes of S(t).We call spine the process S(.).

In fact, given T T t , one can recover (F (t) S(t)).Moreover, with the whole process ( T T t ) t≥0 one can a.s.recover V :
V = t≥0 I S(t) .
As a consequence of the general theory of homogeneous fragmentations (see Bertoin [7]) or by a direct computation, we see that (s(t), t ≥ 0) is an homogeneous Poisson process with parameter 1.In particular, if E(t, z) := (2z) s(t) e t (1−2z)  (60)

then E

(t, z) = 1.Conditionally on F r = F r ∨ σ(
(r), s ≤ r), the restriction of the fragmentation F (. + r) to the interval I S(r) is distributed as a rescaling of F (.) by a factor 2 −s(r) , which entails that E(t, z), F t t≥0 is a martingale.By the size biasing scheme (59) and the definition ( 16) we get
M (t, z) E [E(t, z) | F t ] .(61)
Hence, the Yule martingale appears to be a projection of the martingale E (which is a spinemeasurable function) on the σ-algebra containing only the underlying binary tree.Coming back to the discrete time, set Spine n := S(τ n ) and s n := |Spine n |.Notice that the underlying unm ying (59) at the (F t , t ≥ 0) stopping time τ n , we get for every leaf u ∈ ∂T n (and k ≥ 1) :
P(Spine n = u | F (n) ) = 2 −|u| ,(62)P(s n = k | F (n) ) = U k (n)2 −|k| .
Thus, for fixed n, to draw at random the marked tree T T τn , one may choose at first a binary tree T n , and then pick the marked leaf according to the conditional distribution (62).Let F (0) be the trivial σ-algebra, and for n ≥ 1 let F (n) be the σ-algebra obtained from F (n) by adjunction of S(τ 1 ), ..., S(τ n ).Let us consider E n (z) := E E(τ n , z) | F (n) (with E 0 (z) := 1).From Lemma 2.1 a)

we have E(e τn(1−2z) ) = C n (z) −1 hence
E n (z) = (2z) s n C n (z) −1 . (63)
From the martingale property of E(t, z) and the definition of E n (z) e see that E n (z), F (n) is a martingale.Like in (61), we get easily
M n (z) = E E n (z) | F (n) ,(64)
so that the martingale n (z) are obtained from the "exponential martingales" E(z, t) and E n (z) by projection.Moreover the martingale connection ( 21) may be seen as the projection on F (n) of the relation
E(τ n , z) = n (z)E n (z) .
(z) as
M n (z) = E(M (τ n , z)|F (n) );
this is a kind of integration with respect to the time.All these martingales are precisely the main tool to tilt probabilities.In particular we define P (2z) on ( F t , t ≥ 0) by
P (2z) | F t = E(t, z) P | F t ,(65)
By projection on (F t , t ≥ 0), (65) yields
P (2z) | F t sp.d P (2z) ) is the restriction of P (resp.P (2z) ) to ∨ n F (n) , the discrete versions of the above relations are
d P (2z) | F (n) = E n (z) d P | F (n) , d P (2z) | F (n) = M n (z) d P | F (n) .(67)
It turns out that P (2z) can be seen as a probability on marked Yule trees.This is the object of the following subsectio .


A biased Yule tree

Recall the construction of the Yule tree process (T T t ) t≥0 given in Secti At its death, u becomes an internal node, and two leaves u0 and u1 appear (with new Exp(1), independent of the other ones).

Let us consider now a model of marked binary tree (T T ⋆ t ) t≥0 defined as follows.In T T ⋆ t there are now two kinds of nodes: marked and unmarked.We denote by (v, m) the node v if it is marked, and by (v, m) the node v if it is unmarked.At time 0, T T ⋆ 0 = {(∅, m)}.Each unmarke leaf owns a Exp(1)-distributed clock.The d clock.Now the evolution of the tilted Yule tree is as follows:

• when an unmarked leaf u dies, u becomes an unmarked internal node, and two m) appear.

Once again, the BST can be decomposed along the marked branch.The speed of growing of the marked branch depends on the value of 2z.One may also interpret the size of the subtrees rooted on the tilted branch as t t (see Barbour & al. [2], Pitman [34]), a tions of the behavior of the size of the subtrees rooted on the marked branch.

As in the previous subsection, we denote by (v, m) a marked node and (v, m) an unmarked node.The dyna llowing conditional probabilities: if (v m) ne n , T n+1 = T n ∪ {(v0, m), (v1, m)}| T n ) = 1 n + 2z If (v, m) ∈ ∂ T n , (i.e. Spine n = v), then Q (2z) (Spine n+1 = v0, T 1, T n+1 = T n ∪ {(v0, m), (v1, m)}| T n ) = 1 2 2z n + 2z
.

Summing up, we have for any marked tree t n+1 with n + 1 nodes

hat can be obtained
from T n by one insertion
Q (2z) ( T n+1 = t n+1 | T n ) = z s n+1 −sn n + 2z(68)
and
Q (1) ( T n+1 = t n+1 | T n ) = (1/2) s n+1 −sn n + 1 .
Thus, by iterative construction,
Q (2z) Q (1) Fn = n−1 j=0 (2z) s j+1 −s j (j + 1) j + 2z = (2z) s n C n (z) −1 = E n (z) .
Hence, Q (2z is absolutely continuous with respect to Q (1) , with the Radon-Nikodym derivative announced in (67).Since Q (1) and d IP (1) (the non-biased models) are identical, the law of (
T n ) n under Q (2z) is d IP (2z) .
One finds an analogous result (in another context) and its proof in Lemma 1 and 2 of [14].


Spine evolution

Thanks to the previous subsections, it appears that under d P (2z)
s n = 1 + n−1 1 ǫ k(69)
where (ǫ k ) k≥ are independent and for every k ≥ 1, ǫ k is a Bernoulli random variable with parameter 3) (large deviations) The famil of distributions of (s n , n > 0) under d P (2z) satisfies the large deviation principle on [0, ∞) with speed log n and rate function η 2z where the function η λ is defined in (4).

Proof: 1) and 2) are consequences of known results on sums of independent r.v.(see [33]).Notice also that s n − E (2z) (s n ) is a martingale.

3) is a consequence of Gärtner-Ellis theorem.

Once again, this proposition shows that under the biased model, the BST evolves rather differently that under the usual model.For example, the marked leaf depth is about 2z log n.So, for z > z + c , the marked leaf is higher that the height of the non-biased BST.


Depth of insertion

In introducing the BST model, we defined the sequence (D n , n ≥ 0) as the successive inserted nodes and d n = |D n | (see (1)).In continuous time, we set η(t) = inf{s > t : T T s = T T t } for the first time of growing after t, and D(t) = T T η(t) \ T T t for the ertion.

Let us stress on the difference between the spine processes (s n , n ≥ 0) and (s(t), t ≥ 0) and the insertion processes (d n , n ≥ 0) and (d(t), t ≥ 0).

The (marginal) distribution of d n is given in Jabbour [25] (see also Mahmoud [31])
Ez dn = C n (z) n + 1 =(
Note that (iii) of course, implies that (ii) is not an almost sure convergence.Proof: The arguments to prove (i) and (ii) are classical; (iii) is a consequence of (3).

For the Yule tree, we did not find the distribution of d(t) in the literature.Let us give the joint distribution of (N t , d(t)) (for t fixed).

Since {N t = n + 1} = {τ n ≤ t < τ n+1 }, we have E(z d(t) s Nt ) = ∞ 0 (Ez dn )P(N t = n + 1)s n+1 .Since the distribution of N t is geometric of parameter e −t , and owing to (71) we get

Taking s = 1, we get the marginal of d(t)

Ez d(t) = e t(2z−1) − 1 (e t − 1)(2z − 1)

.

Transforming these generating functions into Fourier transforms, it is now easy to conclude that Proposition 5.3 As t → ∞,
N t e −t , d(t) − 2t √ 2t law =⇒ (ξ, G)
where ξ is defined in (19) and G is N (0, 1) and in ependent of ξ.

Remark: For the same reasons as in (74), we have a.s.lim inf Under the change of probability P (2z) (or using Kolmogorov equations) the distribution of N t is given by: E (2z) φ Nt = E (2z) s(t) e t(1−2z) φ Nt (80
)
where φ is any real in [0, 1].Hence, under P (2z) , the r.v.N t − 1 is a negative binomial of order 2z and parameter e −t .As t → ∞, the P (2z) distribution of e −t N t converges to a γ(2z)-distributed random variable.Actually we have for every z, t, h
E[N t+h |F t ] = (N t − 1)EN h + E (2z) N h(81)
= e h (N t − 1)

of the marked b
anch depends on the value of 2z.If 2z > 1 then, the growing of the than the other ones, when 2z < 1, the growing of the marked branch is slower.The depth of the marked leaf follows a Poisson process of rate 2z.Notice that we have already met this Poisson process in

e proof of Theor
m 4.1.It turns out that under P (2z) , the process ( T T t ) t≥0 has the same distribution as the process (T T ⋆ t ) t≥0 (consider the spine as the marked leaf).For the underlying b

(.)) under P (2z) (
efined by (65)) is the law of (F ⋆ (.), S ⋆ (.)).It follows that, under P (2z) one may also build the spine by choosing at first a uniform random variable V and follow the fragment containing V .This is not true in general when using the tilting method.Usually, at each splitting of the marked fr gment M , one has to choose the new marked fragment among the children of M , according to a rule depending on the size of these fragments.It cannot be s mmed up by the drawing of a random variable V , once for all as in our case, where s zes are equal.According to the representation by (T T ⋆ t ) t≥0 , the Yule tre tion according to the marked branch.Let u be a node of the marked branch.On of the nodes u0 or u1 does not belong to this marked branch.Assume that it is u0.Then, (up to a change of the time origin),• the subtree rooted in u0 is a copy of the u tilted Yule tree;• the subtree rooted in u1 is a copy of the tilted Yule tree.We can also see this process as a branching process with immigratio , as presented in[40](see also[34]chap.10 and[21]).A biased BST modelThe tilted Yule tree can also been stopped at time τ n of the creation time of the nth internal node.Let T n be the obtained marked binary search tree.The discrete evolution is as follows: T n is a complete binary and the n other ones are unmarked.Knowing n , the marked tree T n+1 is as follows: e hoose the marked leaf with probability 2z/(n + 2z) and each unmarked one with probability 1/(n + internal node and two unmarked leaves v0 and v1 are created.• If t e chosen leaf v is marked, v becomes a marked internal node.Two leaves v0 and v1 appear.One marks at random v0 or v1 (equally likely) and let the ot on the marked binary earch tree process ( T n ) n under this model of evolution.
A diffusion limit for a class of randomly-growing binary trees. D Aldous, P Shields, Probab. Theory Related Fields. 791988

Logarithmic combinatorial structures: a probabilistic approach. R Arratia matics. 2003European Mathematical So