  In a recent paper, the question of determining the fraction of binary trees
that contain a fixed pattern known as the snowflake was posed. We show that
this fraction goes to 1, providing two very different proofs: a purely
combinatorial one that is quantitative and specific to this problem; and a
proof using branching process techniques that is less explicit, but also much
more general, as it applies to any fixed patterns and can be extended to other
trees and networks. In particular, it follows immediately from our second proof
that the fraction of $d$-ary trees (resp. level-$k$ networks) that contain a
fixed $d$-ary tree (resp. level-$k$ network) tends to $1$ as the number of
leaves grows.
