trees
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a tree is a widely used abstract data type that represents a hierarchical structure with a set of connected nodes.
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each node in the tree can be connected to many children, but must be connect to exactly one parent (except for the root node).
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a tree is an undirected and connected acyclic graph and there are no cycle or loops.
binary trees
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binary trees are trees that have each up to 2 children.
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access, search, remove, insert are all
O(log(N). space complexity of traversing balanced trees isO(h)wherehis the height of the tree (while very skewed trees will beO(N). -
the width is the number of nodes in a level.
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the degree is the nunber of children of a node.
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a complete tree is a tree on which every level is fully filled (except perhaps for the last).
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a perfect tree is both full and complete (it must have exactly
2**k - 1nodes, wherekis the number of levels).
full trees
- a full binary tree has each node with either zero or two children (and no node has only one child).
def is_full(node) -> bool:
if node is None:
return True
return bool(node.right and node.left) and is_full(node.right) and is_full(node.left)
is leaf?
- a node is called leaf if it has no children.
def is_leaf(node):
return bool(not node.right and not node.left)
depth of a binary tree
- the depth (or level) of node is the number of edges from the tree's root node until the node.
def max_depth(root) -> int:
if root is None:
return 0
return max(max_depth(root.left) + 1, max_depth(root.right) + 1)
height of a tree
-
the height of a node is the number of edges on the longest path between that node and a leaf.
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the height of tree is the height of its root node, or the depth of its deepest node.
def height(root):
if root is none:
return 0
return 1 + max(height(root.left), height(root.right))
balanced trees
- a balanced tree is a binary tree in which the left and right subtrees of every node differ in height by no more than 1.
def height(root):
if root is None:
return -1
return 1 + max(height(root.left), height(root.right))
def is_balanced(root):
if root is None:
return True
return abs(height(root.left) - height(root.right)) < 2 and \
is_balanced(root.left) and is_balanced(root.right)
tree traversal: breath-first search (level-order)
-
give you all elements in order with time
O(log(N). used to traverse a tree by level. -
iterative solutions use a queue for traversal or find the shortest path from the root node to the target node:
- in the first round, we process the root node.
- in the second round, we process the nodes next to the root node. - in the third round, we process the nodes which are two steps from the root node, etc. - newly-added nodes will not be traversed immediately but will be processed in the next round.
- if node X is added to the kth round queue, the shortest path between the root node and X is exactly k.
- the processing order of the nodes in the exact same order as how they were added to the queue (which is FIFO).
def bfs_iterative(root):
result = []
queue = collections.deque([root])
while queue:
node = queue.popleft()
if node:
result.append(node.val)
queue.append(node.left)
queue.append(node.right)
return result
tree traversal: depth-first search
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deep-first search (DFS) can also be used to find the path from the root node to the target node if you want to visit every node and/or search the deepest paths firsts.
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recursion solutions are easier to implement; however, if the recursion depth is too high, stack overflow might occur. in that case, you might use BFS instead or implement DFS using an explicit stack (i.e., use a while loop and a stack structure to simulate the system call stack).
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overall, we only trace back and try another path after we reach the deepest node. as a result, the first path you find in DFS is not always the shortest path:
- we first push the root node to the stack, then we try the first neighbor and push its node to the stack, etc.
- when we reach the deepest node, we need to trace back.
- when we track back, we pop the deepest node from the stack, which is actually the last node pushed to the stack.
- the processing order of the nodes is exactly the opposite order of how they are added to the stack.
in-order
-
left -> node -> right -
in a bst, in-order traversal will be sorted in the ascending order (therefore, it's the most frequently used method).
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converting a sorted array to a bst with inorder has no unique solution (in another hadnd, both preorder and postorder are unique identifiers of a bst).
def inorder(root):
if root is None:
return []
return inorder(root.left) + [root.val] + inorder(root.right)
def inorder_iterative(root) -> list:
result = []
stack = []
node = root
while stack or node:
if node:
stack.append(node)
node = node.left
else:
node = stack.pop()
result.append(node.val)
node = node.right
return result
- we can also build an interator:
class BST_Iterator:
def __init__(self, root):
self.stack = []
self.left_inorder(root)
def left_inorder(self, root):
while root:
self.stack.append(root)
root = root.left
def next(self) -> int:
top_node = self.stack.pop()
if top_node.right:
self.left_inorder(top_node.right)
return top_node.val
def has_next(self) -> bool:
return len(self.stack) > 0
pre-order
-
node -> left -> right -
top-down (parameters are passed down to children), so deserialize with a queue.
def preorder_recursive(root):
if root is None:
return []
return [root.val] + preorder(root.left) + preorder(root.right)
def preorder_iterative(root) -> list:
result = []
stack = [root]
while stack:
node = stack.pop()
if node:
result.append(node.val)
stack.append(node.right) # not the order (stacks are fifo)
stack.append(node.left)
return result
post-order
-
left -> right -> node -
bottom-up solution.
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deletion process is always post-order: when you delete a node, you will delete its left child and its right child before you delete the node itself.
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post-order can be used in mathematical expressions as it's easier to write a program to parse a post-order expression. using a stack, each time when you meet an operator, you can just pop 2 elements from the stack, calculate the result and push the result back into the stack.
def postorder(root):
if root is None:
return []
return postorder(root.left) + postorder(root.right) + [root.val]
def postorder_iterative(root) -> list:
stack, result = [], []
node = root
while node or stack:
while node:
if node.right:
stack.append(node.right)
stack.append(node)
node = node.left
node = stack.pop()
if stack and node.right == stack[-1]:
stack[-1] = node
node = node.right
else:
result.append(node.val)
node = None
return result
is same tree?
def is_same_trees(p, q):
if not p and not q:
return True
if (not p and q) or (not q and p):
return False
if p.val != q.val:
return False
return is_same_trees(p.right, q.right) and is_same_trees(p.left, q.left)
is symmetric?
def is_symmetric(root) -> bool:
stack = [(root, root)]
while stack:
node1, node2 = stack.pop()
if (not node1 and node2) or (not node2 and node1):
return False
if node1 and node2:
if node1.val != node2.val:
return False
stack.append([node1.left, node2.right])
stack.append([node1.right, node2.left])
return True
def is_symmetric_recursive(root) -> bool:
def helper(node1, node2):
if (not node1 and node2) or \
(not node2 and node1) or \
(node1 and node2 and node1.val != node2.val):
return False
if (not node1 and not node2):
return True
return helper(node1.left, node2.right) and helper(node2.left, node1.right)
return helper(root.left, root.right)
lowest common ancestor
def lowest_common_ancestor(root, p, q):
stack = [root]
parent = {root: None}
while p not in parent or q not in parent:
node = stack.pop()
if node:
parent[node.left] = node
parent[node.right] = node
stack.append(node.left)
stack.append(node.right)
ancestors = set()
while p:
ancestors.add(p)
p = parent[p]
while q not in ancestors:
q = parent[q]
return q
has path sum?
def has_path_sum(root, target_sum) -> bool:
def transverse(node, sum_here=0):
if not node:
return sum_here == target_sum
sum_here += node.val
if not node.left:
return transverse(node.right, sum_here)
if not node.right:
return transverse(node.left, sum_here)
else:
return transverse(node.left, sum_here) or transverse(node.right, sum_here)
if not root:
return False
return transverse(root)
build tree from inorder with preorder or postorder
- building with preorder:
def build_tree(preorder, inorder) -> Optional[Node]:
def helper(left, right, index_map):
if left > right:
return None
root = Node(preorder.pop(0)) # this order change from postorder
index_here = index_map[root.val]
root.left = helper(left, index_here - 1, index_map) # this order change from postorder
root.right = helper(index_here + 1, right, index_map)
return root
index_map = {value: i for i, value in enumerate(inorder)}
return helper(0, len(inorder) - 1, index_map)
- build with postorder:
def build_tree(left, right, index_map):
if left > right:
return None
root = Node(postorder.pop()) # this order change from preorder
index_here = index_map[root.val]
root.right = build_tree(index_here + 1, right, index_map) # this order change from preorder
root.left = build_tree(left, index_here - 1, index_map)
return root
def build_tree(inorder, postorder) -> Optional[Node]:
index_map = {val: i for i, value in enumerate(inorder)}
return fill_tree(0, len(inorder) - 1, index_map)
return number of unival subtrees
- a unival subtree means all nodes of the subtree have the same value
def count_unival(root) -> int:
global count = 0
def dfs(node):
if node is None:
return True
if dfs(node.left) and dfs(node.right):
if (node.left and node.left.val != node.val) or \
(node.right and node.right.val != node.val):
return False
self.count += 1
return True
return False
dfs(root)
return count
successors and precessors
def successor(root):
root = root.right
while root.left:
root = root.left
return root
def predecessor(root):
root = root.left
while root.right:
root = root.right
return root
binary search trees
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binary search tree are binary trees where all nodes on the left are smaller than the root, which is smaller than all nodes on the right.
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if a bst is balanced, it guarantees
O(log(N))for insert and search (as we keep the tree's height ash = log(N)). -
common types of balanced trees are red-black and avl.
insert a node
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the main strategy is to find out a proper leaf position for the target and then insert the node as a leaf (therefore, insertion will begin as a search).
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the time complexity is
O(h)wherehis a tree height. that results inO(log(N))in the average case, andO(N)worst case.
def bst_insert_iterative(root, val):
node = root
while node:
if val > node.val:
if not node.right:
node.right = Node(val)
break
else:
node = node.right
else:
if not node.left:
node.left = Node(val)
break
else:
node = node.left
return root
def bst_insert_recursive(root, val):
if root is None:
return Node(val)
if val > root.val:
root.right = self.bst_insert_recursive(root.right, val)
else:
root.left = self.bst_insert_recursive(root.left, val)
return root
delete a node
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deletion is a more complicated operation, and there are several strategies.
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one of them is to replace the target node with a proper child: - if the target node has no child (it's a leaf): simply remove the node - if the target node has one child, use the child to replace the node - if the target node has two child, replace the node with its in-order successor or predecessor node and delete the node
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similar to the recursion solution of the search operation, the time complexity is
O(h)in the worst case. -
according to the depth of recursion, the space complexity is also
O(h)in the worst case. we can also represent the complexity using the total number of nodesN. -
the time complexity and space complexity will be
O(log(N))in the best case butO(N)in the worse case.
def successor(root):
root = root.right
while root.left:
root = root.left
return root.val
def predecessor(root):
root = root.left
while root.right:
root = root.right
return root.val
def delete_node(root, key):
if root is None:
return root
if key > root.val:
root.right = delete_node(root.right, key)
elif key < root.val:
root.left = delete_node(root.left, key)
else:
if not (root.left or root.right):
root = None
elif root.right:
root.val = successor(root)
root.right = delete_node(root.right, root.val)
else:
root.val = predecessor(root)
root.left = delete_node(root.left, root.val)
return root
search for a value
-
for the recursive solution, in the worst case, the depth of the recursion is equal to the height of the tree. therefore, the time complexity would be
O(h). the space complexity is alsoO(h). -
for an iterative solution, the time complexity is equal to the loop time which is also
O(h), while the space complexity isO(1).
def search_bst_recursive(root, val):
if root is None or root.val == val:
return root
if val > root.val:
return search_bst_recursive(root.right, val)
else:
return search_bst_recursive(root.left, val)
def search_bst_iterative(root, val):
while root:
if root.val == val:
break
if root.val < val:
root = root.right
else:
root = root.left
return root
find successor of two nodes inorder
def find_successor(node1, node2):
successor = None
while node1:
if node1.val <= node2.val:
node1 = node1.right
else:
successor = node1
node1 = node1.left
return successor
convert sorted array to bst
- note that there is no unique solution.
def convert_sorted_array_to_bst(nums):
def helper(left, right):
if left > right:
return None
p = (left + right) // 2
root = Node(nums[p])
root.left = helper(left, p - 1)
root.right = helper(p + 1, right)
return root
return helper(0, len(nums) - 1)
lowest common ancestor for a bst
def lowest_common_ancestor(root, p, q):
node, result = root, root
while node:
result = node
if node.val > p.val and node.val > q.val:
node = node.left
elif node.val < p.val and node.val < q.val:
node = node.right
else:
break
return result
checking if bst is valid
def is_valid_bst_iterative(root):
queue = deque((root, float(-inf), float(inf)))
while queue:
node, min_val, max_val = queue.popleft()
if node:
if min_val >= node.val or node.val >= max_val:
return False
if node.left:
queue.append((node.left, min_val, node.val))
if node.right:
queue.append((node.right, node.val, max_val))
return True
def is_valid_bst_recursive(root, min_val=float(-inf), max_val=float(inf)):
if root is None:
return True
return (min_val < root.val < max_val) and \
is_valid_bst_recursive(root.left, min_val, root.val) and \
is_valid_bst_recursive(root.right, root.val, max_val)
def is_valid_bst_inorder(root):
def inorder(node):
if node is None:
return True
inorder(node.left)
stack.append(node.val)
inorder(node.right)
stack = []
inorder(root)
for i in range(1, len(stack)):
if queue[i] <= queue[i - 1]:
return False
return True