[mypy] Fix type annotations in wavelet_tree.py (#5641)

* [mypy] Fix type annotations for wavelet_tree.py

* fix a typo

data_structures/binary_tree/wavelet_tree.py

@@ -31,7 +31,7 @@ class Node:
         return f"min_value: {self.minn}, max_value: {self.maxx}"
 
 
-def build_tree(arr: list[int]) -> Node:
+def build_tree(arr: list[int]) -> Node | None:
     """
     Builds the tree for arr and returns the root
     of the constructed tree
@@ -51,7 +51,10 @@ def build_tree(arr: list[int]) -> Node:
     then recursively build trees for left_arr and right_arr
     """
     pivot = (root.minn + root.maxx) // 2
-    left_arr, right_arr = [], []
+
+    left_arr: list[int] = []
+    right_arr: list[int] = []
+
     for index, num in enumerate(arr):
         if num <= pivot:
             left_arr.append(num)
@@ -63,7 +66,7 @@ def build_tree(arr: list[int]) -> Node:
     return root
 
 
-def rank_till_index(node: Node, num: int, index: int) -> int:
+def rank_till_index(node: Node | None, num: int, index: int) -> int:
     """
     Returns the number of occurrences of num in interval [0, index] in the list
 
@@ -79,7 +82,7 @@ def rank_till_index(node: Node, num: int, index: int) -> int:
     >>> rank_till_index(root, 0, 9)
     1
     """
-    if index < 0:
+    if index < 0 or node is None:
         return 0
     # Leaf node cases
     if node.minn == node.maxx:
@@ -93,7 +96,7 @@ def rank_till_index(node: Node, num: int, index: int) -> int:
         return rank_till_index(node.right, num, index - node.map_left[index])
 
 
-def rank(node: Node, num: int, start: int, end: int) -> int:
+def rank(node: Node | None, num: int, start: int, end: int) -> int:
     """
     Returns the number of occurrences of num in interval [start, end] in the list
 
@@ -114,7 +117,7 @@ def rank(node: Node, num: int, start: int, end: int) -> int:
     return rank_till_end - rank_before_start
 
 
-def quantile(node: Node, index: int, start: int, end: int) -> int:
+def quantile(node: Node | None, index: int, start: int, end: int) -> int:
     """
     Returns the index'th smallest element in interval [start, end] in the list
     index is 0-indexed
@@ -129,7 +132,7 @@ def quantile(node: Node, index: int, start: int, end: int) -> int:
     >>> quantile(root, 4, 2, 5)
     -1
     """
-    if index > (end - start) or start > end:
+    if index > (end - start) or start > end or node is None:
         return -1
     # Leaf node case
     if node.minn == node.maxx:
@@ -155,10 +158,10 @@ def quantile(node: Node, index: int, start: int, end: int) -> int:
 
 
 def range_counting(
-    node: Node, start: int, end: int, start_num: int, end_num: int
+    node: Node | None, start: int, end: int, start_num: int, end_num: int
 ) -> int:
     """
-    Returns the number of elememts in range [start_num, end_num]
+    Returns the number of elements in range [start_num, end_num]
     in interval [start, end] in the list
 
     >>> root = build_tree(test_array)
@@ -175,6 +178,7 @@ def range_counting(
     """
     if (
         start > end
+        or node is None
         or start_num > end_num
         or node.minn > end_num
         or node.maxx < start_num