Getting to the root of the problem: sums of squares for limits of trees.

UNLABELLED: 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, Székely 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 in a sequence of trees of growing sizes. Finally, we are able to prove the non-convexity of some of these profiles. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s10107-025-02305-1.