On the number of t-ary trees with a given path length
<|reference_start|>On the number of t-ary trees with a given path length: We show that the number of $t$-ary trees with path length equal to $p$ is $\exp(h(t^{-1})t\log t \frac{p}{\log p}(1+o(1)))$, where $\entropy(x){=}{-}x\log x {-}(1{-}x)\log (1{-}x)$ is the binary entropy function. Besides its intrinsic combinatorial interest, the question recently arose in the context of information theory, where the number of $t$-ary trees with path length $p$ estimates the number of universal types, or, equivalently, the number of different possible Lempel-Ziv'78 dictionaries for sequences of length $p$ over an alphabet of size $t$.<|reference_end|>
@article{seroussi2005on,
    title={On the number of t-ary trees with a given path length},
    author={Gadiel Seroussi},
    journal={Algorithmica, Vol. 46, No. 3, pp. 557--565, 2006},
    year={2005},
    doi={10.1007/s00453-006-0122-8},
    archivePrefix={arXiv},
    eprint={cs/0509046},
    primaryClass={cs.DM cs.IT math.IT}
}