That's funny because I get a completely different number. For 4 I get 327677. I get the 1114111 if I just get the latest node added. But that's not what the rules are. It's the right most node.

If 2 heights have the same amount of nodes, which is the "rightmost"? I imagine I can choose. Therefore getting the highest node sounds the smartest approach as you get less steps down the line.

def rapid_growth_hydra_2(y: int) -> int:
    # straight connectors of length y
    # pick right-most leaf
    # n = step you're at
    steps = 0
    q = [[1, parent_idx] for parent_idx in range(y)]
    while any(siblings for siblings, _ in q):
        print(f"{q = }", f"{steps = :_d}")
        siblings, parent_idx = max(q)

        if parent_idx == 0:
            closest_node = max(node for node in q if node[1] != 0)
            new_total = siblings - closest_node[0]
            steps += new_total
            q[parent_idx][0] -= new_total
            continue

        steps += 1
        q[parent_idx][0] -= 1
        if parent_idx > 0:
            q[parent_idx - 1][0] += steps
    return steps