By embedding in a suitable continuous-time process, we find a strong law for $h_n$, the height of a random binary pyramid of order $n$. We show that $h_n/\ln n$ converges almost surely to a constant limit and we determine that limit.
"A Strong Law for the Height of Random Binary Pyramids." Ann. Appl. Probab. 4 (3) 923 - 932, August, 1994. https://doi.org/10.1214/aoap/1177004977