This article, and and "Reconstructing the Tree of Life", inspired me to construct Fibonacci and Lucas rooted binary trees along with their distance matrices. For Fibonacci start off with a root A, representing a species if you like or an individual organism. From it emanate two leaves B and C. In turn B gives rise to two new leaves, D and E, but from C only emanates one, F. So three leaves emerge at this stage. From D emanates two new leaves, G and H; from E only one, namely I; but from F also emanate two, J and K. So five leaves at this stage. And so on. The general rule is that if a given leaf X only produces one leaf then that daughter leaf must go on to produce two, but if X produces two, then only one of those can itself can produce two and the other just one. This rule also applies to Lucas except that we begin with two roots which fuse into one leaf, which in turn splits into three. Of those three, one produces two leaves, but the other two only one each - so four altogether. This could model the numerical difference some kind of sexual (individual or inter-species) reproduction makes. In both cases we can have the species or individual die out after its act of evolution or reproduction, or remain in existence, as the numbers given only refer to new leaves.
Some very interesting distance matrices emerge. In my version of the Fibonacci, it's B followed by A which has the lowest value by my reckoning if we assume a value of 1 for every edge.