The semi-random tree process Burova, Sofiya Lichev, Lyuben The online semi-random graph process is a one-player game which starts with the empty graph on n vertices. At every round, a player (called Builder) is presented with a vertex v chosen uniformly at random and independently from previous rounds, and constructs an edge of their choice that is incident to v. Inspired by recent advances on the semi-random graph process, we define a family of generalized online semi-random models. We analyse a particular instance that shares similar features with the original semi-random graph process and determine the hitting times of the classical graph properties minimum degree k,k-connectivity, containment of a perfect matching, a Hamiltonian cycle and an H-factor for a fixed graph H possessing an additional tree-like property. Along the way, we derive a few consequences of the famous Aldous-Broder algorithm that may be of independent interest. Elsevier 2025 info:eu-repo/semantics/article doc-type:Article Article http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/19018 Burova S, Lichev L. The semi-random tree process. <i>European Journal of Combinatorics</i>. 2025;126. doi:<a href="https://doi.org/10.1016/j.ejc.2025.104120">10.1016/j.ejc.2025.104120</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1016/j.ejc.2025.104120 info:eu-repo/semantics/altIdentifier/issn/0195-6698 info:eu-repo/semantics/altIdentifier/wos/001420659400001 info:eu-repo/semantics/altIdentifier/arxiv/2204.07376 info:eu-repo/semantics/openAccess