We only consider

Encode tree-limits as sequences of

$(\#$$,\#$$,\#$$,\#$$,\ldots)$.

And normalize to **densities**

$(d_{\mathcal{T}}($$),d_{\mathcal{T}}($$),d_{\mathcal{T}}($$),d_{\mathcal{T}}($$),\ldots),$

where $$\sum_{\text{tree }T \text{ with $k$ vertices}} d_{\mathcal{T}}(T) = 1. $$

$(d_{\mathcal{T}}($$),d_{\mathcal{T}}($$),d_{\mathcal{T}}($$),d_{\mathcal{T}}($$),\ldots),$

where $$\sum_{\text{tree }T \text{ with $k$ vertices}} d_{\mathcal{T}}(T) = 1. $$

What happens, if we only consider the **leaves** of trees to be the "vertices" of the tree?

(Inner vertices are now part of the "edges" of the tree.)

**Motivation:** phylogenetic trees, tanglegrams

$$\longrightarrow$$

$$\longrightarrow$$

$$\longrightarrow$$

Let $\mathcal{T} = (T_i)_{i\geq 0}$ be a tree limit. Let $S$ be a finite tree. $$d_{\mathcal{T}}({\color{darkorange}S}):=\lim_{i\to\infty} \mathbb{P}[\left.(T_i)\right|_{ {\color{green}V_i}} \cong {\color{darkorange}S}],$$ where $\color{green}V_i$ is a random subset of leaves of $T_i$ of size $V({\color{darkorange}S})$.

$(d_{\mathcal{T}}($$),d_{\mathcal{T}}($$),d_{\mathcal{T}}($$),d_{\mathcal{T}}($$),\ldots)$

$\hat{=}$ Dirac measure of tree $\mathcal{T}$

$\xrightarrow{Dualize}$ Nonnegative functions on tree-limits

$\longrightarrow$ Flag sums-of-squares

$\longrightarrow$ **Semidefinite programming!**