On the enumeration of tanglegrams and tangled chains

Tanglegrams are a special class of graphs appearing in applications concerning cospeciation and coevolution in biology and computer science. They are formed by identifying the leaves of two rooted binary trees. We give an explicit formula to count the number of distinct binary rooted tanglegrams with $n$ matched vertices, along with a simple asymptotic formula and an algorithm for choosing a tanglegram uniformly at random. The enumeration formula is then extended to count the number of tangled chains of binary trees of any length. This includes a new formula for the number of binary trees with $n$ leaves. We also give a conjecture for the expected number of cherries in a large randomly chosen binary tree and an extension of this conjecture to other types of trees.