Given n, how many structurally unique BST's (binary search trees) that store values 1 ... n?

Example:


Input: 3
Output: 5
Explanation:
Given n = 3, there are a total of 5 unique BST's:

   1         3     3      2      1
    \       /     /      / \      \
     3     2     1      1   3      2
    /     /       \                 \
   2     1         2                 3

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'class Solution:\n     def numTrees(self, n):\n         """\n         :type n: int\n         :rtype: int\n         """\n         \n         """\n         The most straightforward solution would be to try all possible permutations that specify the order of input in constructing a BST and counting the different ways in which the BST results. But that would take O(n!) time and O(n) space.\n         Maybe it would help to use the fact that a BST is defined recursively.\n         Does the topology of a subtree affects whether the parent subtree is valid or not?\n         No.\n         And is it possible that we get any duplicate trees when we first designate our root and then permute its subtrees? \n         No.\n         So for each node i, set i as a root and then construct/permute the left and the right subtrees. \n         The parameter we need to take care of is the list of available nodes in that subtree.\n         Memoize the results, too.\n         """\n         \n         def memoize(func):\n             func.cache = cache = dict()\n             def memoizer(*args):\n                 key = str(args)\n                 if key not in cache:\n                     cache[key] = func(*args)\n                 return cache[key]\n             return memoizer\n                 \n         \n         @memoize\n         def count(root, lower_bound, upper_bound):\n             \n             if upper_bound < lower_bound:\n                 return 0\n             \n             if lower_bound == upper_bound:\n                 return 1\n                 \n             left_combinations = max(1, sum(count(left, lower_bound, root - 1) for left in range(lower_bound, root)))\n             right_combinations = max(1, sum(count(right, root + 1, upper_bound) for right in range(root + 1, upper_bound + 1)))\n             #print(root, lower_bound, upper_bound, left_combinations, right_combinations)\n             return left_combinations * right_combinations\n         \n         return sum(count(root, 1, n) for root in range(1, n + 1))\n         \n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'numTrees'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 11462