What does it mean to order subrees of binary tree by inclusion?

Question

A leaf labeled binary tree is a tree $T$ whose nodes are of degree 1 or 3, with $n$ leaves labeled as $s_1 , s_2 ,..., s_n$.
Removing an edge $e$ from $T$ results in two subtrees of $T$ - $T_1$ and $T_2$.
What does it mean to order $T$'s subtrees by inclusion, and then compute the linear extension of that order?
Here's a simple binary tree to illustrate the idea on:
Link (I'm having trouble uploading the image directly)

Answer 1

Any collection of sets can be partially ordered by inclusion. The ordering is given by $$A\leq B\iff A\subseteq B$$ A linear extension of a partial order $\leq$ is a total order $\preceq$ such that $$A\leq B\implies A\preceq B$$ Finding all linear extensions is quite a hard thing to do in terms of computability, but in small cases you can use some heuristics to find them all by hand relatively quickly.