  The inducibility of a graph represents its maximum density as an induced
subgraph over all possible sequences of graphs of size growing to infinity.
This invariant of graphs has been extensively studied since its introduction in
$1975$ by Pippenger and Golumbic. In $2017$, Czabarka, Sz\'ekely and Wagner
extended this notion to leaf-labeled rooted binary trees, which are objects
widely studied in the field of phylogenetics. They obtain the first results and
bounds for the densities and inducibilities of such trees. Following up on
their work, we apply Razborov's flag algebra theory to this setting,
introducing the flag algebra of rooted leaf-labeled binary trees. This
framework allows us to use polynomial optimization methods, based on
semidefinite programming, to efficiently obtain new upper bounds for the
inducibility of trees and to improve existing ones. Additionally, we obtain the
first outer approximations of profiles of trees, which represent all possible
simultaneous densities of a pair of trees. Finally, we are able to prove the
non-convexity of some of these profiles.
