Consider a full binary tree with $k>10$ levels. Let the *lengths* of individual edges in this tree be i.i.d. random variables with finite moments. Then total lengths of the $2^{k-1}$ source-to-sink paths in this tree are approximately Gaussian by the CLT, regardless of the edge-length distribution. We are interested in the limit distribution ($k\rightarrow\infty$) of the maximum path length in the tree.

Our numerical simulation built 100K independent trees ($k=15$) with $(a)$ uniform and $(b)$ Gaussian edge lengths. The resulting distributions for $(a)$ and $(b)$ did not look qualitatively different and were somewhat skewed to the right. Lognormal distributions provided very close fits --- better than *Gumbel* and *Airy*. If lognormal is indeed the limit distribution, we would appreciate references or suggestions on proving this analytically.