Tree based optimization problems stand for those paradigms where solutions can be arranged within a tree-like graph whose nodes represent the optimization variables of the problem at hand and their interconnecting edges topological and/or hierarchical relationships between such variables. In this context, a research line of increasing interest during the last decade focuses on the derivation of intelligent solution encoding strategies capable of 1) capturing all topological constraints of this particular class of graphs; and 2) preserving their connectivity properties when they undergo combination/mutation operations within approximative evolutionary solvers. This manuscript takes a step over the state of the art by shedding light on the heri-tability properties of the Dandelion tree encoding approach under avant-garde stochastically-controlled evolutionary operators. In particular we elaborate on the topological heritability of the so-called Harmony Memory Considering Rate (HMCR) exploitative operator of the Harmony Search algorithm, a population-based meta-heuristic algorithm that has so far shown to outperform other evolutionary schemes in a wide range of optimization scenarios. Results from extensive Monte Carlo simulations are discussed in terms of the preserved structural properties of the newly produced solutions with respect to the initial Dandelion-encoded population.