Explore a comprehensive lecture on combinatorial statistics and its applications in various scientific fields. Delve into the intricacies of broadcast on trees, probability and statistics of dependent variables, and their connections to phylogenetic inference in biology. Examine the block model conjecture and its implications, while gaining insights into linearization of belief propagation and non-backtracking random walks. Discover how these concepts relate to the cavity and replica method, and understand the fractal argument through visual representations. Reflect on the depth of data analysis and consider future directions in this fascinating field of study.