Representation of ordered trees with a given degree distribution

The degree distribution of an ordered tree $T$ with $n$ nodes is $\vec{n} = (n_0,\ldots,n_{n-1})$, where $n_i$ is the number of nodes in $T$ with $i$ children. Let $\mathcal{N}(\vec{n})$ be the number of trees with degree distribution $\vec{n}$. We give a data structure that stores an ordered tree $T$ with $n$ nodes and degree distribution $\vec{n}$ using $\log \mathcal{N}(\vec{n})+O(n/\log^t n)$ bits for every constant $t$. The data structure answers tree queries in constant time. This improves the current data structures with lowest space for ordered trees: The structure of Jansson et al.\ [JCSS 2012] that uses $\log\mathcal{N}(\vec{n})+O(n\log\log n/\log n)$ bits, and the structure of Navarro and Sadakane [TALG 2014] that uses $2n+O(n/\log^t n)$ bits for every constant $t$.