The memo that remembered too much

A staircase-paths counter with memoization returns correct answers — the first time. Called again with a *different* set of allowed step sizes, it returns the answer from the previous configuration. A tree helper has the same disease: collect_leaves(tree) returns [2, 3] on the first call and [2, 3, 2, 3] on the second, with the same tree. Each function works in isolation and fails only across calls. Find the bug.

1def count_paths(n, steps, memo={}):
2    """Number of ways to climb n stairs taking any step size in steps."""
3    if n == 0:
4        return 1
5    if n < 0:
6        return 0
7    if n in memo:
8        return memo[n]
9    memo[n] = sum(count_paths(n - s, steps, memo) for s in steps)
10    return memo[n]
11 
12 
13def collect_leaves(node, out=[]):
14    """Leaf values of a binary tree, left to right."""
15    if node is None:
16        return out
17    if node["left"] is None and node["right"] is None:
18        out.append(node["val"])
19        return out
20    collect_leaves(node["left"], out)
21    collect_leaves(node["right"], out)
22    return out
23 
24 
25print(count_paths(4, [1, 2]))   # 5 — correct
26print(count_paths(4, [1]))      # 5 — expected 1: only 1+1+1+1 is possible
27 
28tree = {"val": 1,
29        "left": {"val": 2, "left": None, "right": None},
30        "right": {"val": 3, "left": None, "right": None}}
31print(collect_leaves(tree))     # [2, 3]
32print(collect_leaves(tree))     # [2, 3, 2, 3] — grows on every call

1. When is the expression {} in memo={} evaluated — at each call, or somewhere else? Where does the resulting dict live between calls?
2. Explain the wrong 5: which entries are in memo when the second call starts, and why is keying the memo by n alone already a design bug once steps can vary?
3. Trace the two collect_leaves calls and explain the doubling.
4. Write the standard fix with a None sentinel, and an alternative using functools.lru_cache. What does lru_cache require of the arguments?
5. When is a mutable default actually the intended behaviour, and how would you make that intent unmistakable?
6. State the complexity of the fixed count_paths.