Compacted binary trees admit a stretched exponential

Abstract

A compacted binary tree is a directed acyclic graph encoding a binary tree in which common subtrees are factored and shared, such that they are represented only once. We show that the number of compacted binary trees of size n grows asymptotically likewhere a 1 ≈ −2.338 is the largest root of the Airy function. Our method involves a new two parameter recurrence which yields an algorithm of quadratic arithmetic complexity. We use empirical methods to estimate the values of all terms defined by the recurrence, then we prove by induction that these estimates are sufficiently accurate for large n to determine the asymptotic form. Our results also lead to new bounds on the number of minimal finite automata recognizing a finite language on a binary alphabet. As a consequence, these also exhibit a stretched exponential.

On the Asymptotic Growth of the Number of Tree-Child Networks

“…Similar asymptotic results but for different combinatorial counting problems were obtained by Elvey Price et al in [3], where the (unusual) term exp{cn α } with c some constant and α < 1 was called a quenched exponential. In fact, the method from [3] will also play a crucial role in the proof of our result.…”

“…Now, for m ≈ n, our coefficients become (2n + m − 2)/(2n + m − 3) ≈ 1 and 2n + m − 2 ≈ 3n compared to the coefficients of [3] which become 1 and m+1 ≈ n. Thus, we can expect that the method from [3] will apply to our sequence when divided by 3 n . This is indeed the case.…”

“…The method we are going to use was introduced in [3], where it was first applied to a problem on lattice paths. Indeed, the heuristic for the method comes from lattice path theory and is well-explained in [3].…”

“…Our problem has (so far) nothing to do with lattice paths, however, our recurrence from Corollary 2 has the same shape as the one for which the method was introduced in [3]. The only difference is that the coefficients on the right-hand side are slightly different: in [3] the coefficients were 1 and m + 1, whereas our coefficients are (2n + m − 2)/(2n + m − 3) and 2n + m − 2. But note that we are interested in the asymptotics of b n,n since by (5) and initial conditions:…”

“…is indeed equal to a n−1 . Finally, for a n we will be able to find a recurrence to which the method from [3] can be applied. The above will be done in the next three sections of the paper.…”