Bounds on the Size of the TBR Unit-Neighbourhood

Abstract

In this paper, we study the unit-neighbourhood of the tree bisection and reconnection operation on unrooted binary phylogenetic trees. Specifically, we provide a recursive method to calculate the size of the unit-neighbourhood for any tree in the space T n of unrooted binary phylogenetic trees with n-leaves. We also give both upper and lower bounds on this size for all trees in T n , and characterize those trees for which the stated upper bound is sharp.

On the neighbourhoods of trees

“…x 4 Recall that the exact upper bound on the size of N tbr (T ) for a tree T ∈ T n was proven in [3] by induction on n. Corollary 4.2 confirms this result using a different approach.…”

“…Corollary 4.2 (Theorem 2.1, [3]). The tree T ∈ T n maximises the size of the tbr neighbourhood over T n if and only if T is a caterpillar.…”

“…In Section 4, we characterise the trees that respectively maximise and minimise the size of N tbr (T ) for all binary tree spaces T n . These characterisations are also extended to reprove the tight upper bound given in [3], and to further prove an asymptotically tight lower bound.…”

“…It was also demonstrated in [1] that the size of the tbr neighbourhood is dependent on the shape of T . More recently, in [3], the bounds…”