Generative Structural Theory v1.72 — Unit-Budget Reachability of Trees with Arbitrary Finite Branches at Zero on the Cofinally Complete Growth Sequence E

v1.71 showed that on the sequence E of v1.63 Example Φ2, from the directed triangle y = {(0, 1), (1, 2), (2, 0)} at the stage 0, budget 1 reaches a tree T with node set N_0 through a bidirected tree exactly when T has finitely many edges not incident with 0, and left open the trees with infinitely many such edges all of whose branches at zero are finite. This text shows that budget 1 reaches every tree all of whose branches at zero are finite, with no common bound on their size, and with branching inside the branches and labels not consecutive allowed. The branches, ordered by their least nodes, are B_1, B_2, ..., each with exactly one node joined to 0 (Lemma Q_Ϝ). On a finite node set, from a triangle state X(H; p, q) = P_H ∪ {(0, p), (p, q), (q, 0)} of v1.71 in which every node of a finite tree B, not holding p or q, is joined to 0 and to nothing else, f_B + 3 i_B rounds of the moves of v1.71 Lemma M_Ϟ, each move deleting exactly one pair, attach B as it is in the target and give back a triangle state with the same two core nodes, which need not be leaves of the target (Lemma G_Ϝ; f_B = |B| - 1, i_B the number of nodes of B with a child); preceded by two rounds, the switch, that make two other nodes the core nodes and join the old ones to 0, this is a phase of 2 + f_B + 3 i_B rounds, in which B may hold the old core nodes (Lemma P_Ϝ). On E, phase j starts at the least even stage 2m at or after the end of phase j - 1 with m + 1 larger than every node of B_1, ..., B_j and than the core nodes, takes the auxiliaries m + 1 and m + 2 and assembles B_j among the nodes of V₂ₘ, the new node of each odd stage being hung at 0. Every term is in M_n and holds a pair whose reverse it does not hold, every transition deletes at most one pair, the edges of a finished branch are never removed, every node is a core node or the node pulled at finitely many stages, every pair outside P_T is present at finitely many stages, and L = U = P_T (Proposition S_Ϝ). So, on E, for every tree T with node set N_0 such that every component of T minus {0} is finite, some selection sequence from y at 0 with uniform deletion budget 1, none of whose terms is a bidirected tree, stabilizes with L = U = P_T (Theorem B_Ϝ); with v1.66 Corollary Δ1 (b) and v1.68 Theorem A_Ω, d_min(y; 0, T) = 1 if every branch of T at zero is finite and d_min(y; 0, T) = 2 if some branch is infinite, for every tree T with node set N_0 (Corollary Ϝ1). The statements concern the sequence E and the start y; other starts are left open.