Here, we report the numerical study of time dependent storage of energy by melting a phase change material. The heating is provided along invading lines, which change from single-line invasion to tree-shaped invasion. The numerical simulations show that the history of the amount of melted material is S-shaped. We also found that the fastest melting (i.e., the steepest S curve) is discovered by allowing the tree architecture to morph freely, toward greater access over time. The stem length and branching angle of invading trees can be selected such that the complete melting process is shorter. The melting process can also be made faster by increasing the complexity of the tree structure.