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1995 The distribution of {$3x+1$} trees
David Applegate, Jeffrey C. Lagarias
Experiment. Math. 4(3): 193-209 (1995).


Backwards iteration of the $3x+1$ function starting from a fixed integer a produces a tree of preimages of a. Let $\T_{k} (a)$ denote this tree grown to depth k, and let $\T^*_{k}(a)$ denote the pruned tree resulting from the removal of all nodes $n \equiv 0 \pmod{3}$. We previously computed the maximal and minimal number of leaves in $\T^*_{k} (a)$ for all $a \not\equiv 0 \pmod{3}$ and all $k\le30$. Here we compare these data with predictions made using branching process models designed to imitate the growth of $3x+1$ trees, developed in [Lagarias and Weiss 1992]. We derive rigorous results for the branching process models. The range of variation exhibited by the $3x+1$ trees appears significantly narrower than that of the branching process models. We also study the variation in expected leaf-counts associated to the congruence class of $a \pmod{3^j}$. This variation, when properly normalized, converges almost everywhere as $j \to \infty$ to a limit function on the invertible 3-adic integers.


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David Applegate. Jeffrey C. Lagarias. "The distribution of {$3x+1$} trees." Experiment. Math. 4 (3) 193 - 209, 1995.


Published: 1995
First available in Project Euclid: 3 September 2003

zbMATH: 0868.11012
MathSciNet: MR1387477

Primary: 11B83
Secondary: 60J80

Rights: Copyright © 1995 A K Peters, Ltd.

Vol.4 • No. 3 • 1995
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