Dendric words and morphisms

Given an infinite sequence of symbols (called letters) x, with each pattern w appearing in x, we can associate an extension graph. This graph is a bipartite graph describing the letters that we can see to the left and to the right of w in x. If this graph is a tree, then we say that w is dendric. And if it is the case for all w, we say that x itself is dendric. This notion generalizes the well-studied class of Sturmian words.
A natural question is the preservation of dendricity under a morphism. More precisely, given some x dendric and a morphism, under which conditions is the image of x under that morphism dendric?
Even though the question is still open, we can find some intermediary results using only basic tools from combinatorics on words.

The goal of this talk is to introduce dendric words and present two of these results, namely the evolution of the factor complexity and the description of the morphism preserving dendricity for all dendric words.