In this tutorial, you will learn what a B-tree is. Also, you will find working examples of search operation on a B-tree in Python.
B-tree is a special type of self-balancing search tree in which each node can contain more than one key and can have more than two children. It is a generalized form of the binary search tree.
It is also known as a height-balanced m-way tree.
The need for B-tree arose with the rise in the need for lesser time in accessing the physical storage media like a hard disk. The secondary storage devices are slower with a larger capacity. There was a need for such types of data structures that minimize the disk accesses.
Other data structures such as a binary search tree, avl tree, red-black tree, etc can store only one key in one node. If you have to store a large number of keys, then the height of such trees becomes very large and the access time increases.
However, B-tree can store many keys in a single node and can have multiple child nodes. This decreases the height significantly allowing faster disk accesses.
- For each node x, the keys are stored in increasing order.
- In each node, there is a boolean value x.leaf which is true if x is a leaf.
- If n is the order of the tree, each internal node can contain at most n – 1 keys along with a pointer to each child.
- Each node except root can have at most n children and at least n/2 children.
- All leaves have the same depth (i.e. height-h of the tree).
- The root has at least 2 children and contains a minimum of 1 key.
- If n ≥ 1, then for any n-key B-tree of height h and minimum degree
t ≥ 2,
h ≥ logt (n+1)/2.
Searching for an element in a B-tree is the generalized form of searching an element in a Binary Search Tree. The following steps are followed.
- Starting from the root node, compare k with the first key of the node.
k = the first key of the node, return the node and the index.
k.leaf = true, return NULL (i.e. not found).
k < the first key of the root node, search the left child of this key recursively.
- If there is more than one key in the current node and
k > the first key, compare k with the next key in the node.
k < next key, search the left child of this key (ie. k lies in between the first and the second keys).
Else, search the right child of the key.
- Repeat steps 1 to 4 until the leaf is reached.
- Let us search key
k = 17in the tree below of degree 3.
- k is not found in the root so, compare it with the root key.
k > 11, go to the right child of the root node.
- Compare k with 16. Since
k > 16, compare k with the next key 18.
k < 18, k lies between 16 and 18. Search in the right child of 16 or the left child of 18.
- k is found.
Algorithm for Searching an Element
BtreeSearch(x, k) i = 1 while i ≤ n[x] and k ≥ keyi[x] // n[x] means number of keys in x node do i = i + 1 if i n[x] and k = keyi[x] then return (x, i) if leaf [x] then return NIL else return BtreeSearch(ci[x], k)
To learn more about different B-tree operations, please visit
- Insertion on B-tree
- Deletion on B-tree
/* Searching a key on a B-tree in Python */ /* Create node */ class BTreeNode: def __init__(self, leaf=False): self.leaf = leaf self.keys =  self.child =  class BTree: def __init__(self, t): self.root = BTreeNode(True) self.t = t /* Print the tree */ def print_tree(self, x, l=0): print("Level ", l, " ", len(x.keys), end=":") for i in x.keys: print(i, end=" ") print() l += 1 if len(x.child) > 0: for i in x.child: self.print_tree(i, l) /* Search key */ def search_key(self, k, x=None): if x is not None: i = 0 while i < len(x.keys) and k > x.keys[i]: i += 1 if i < len(x.keys) and k == x.keys[i]: return (x, i) elif x.leaf: return None else: return self.search_key(k, x.child[i]) else: return self.search_key(k, self.root) /* Insert the key */ def insert_key(self, k): root = self.root if len(root.keys) == (2 * self.t) - 1: temp = BTreeNode() self.root = temp temp.child.insert_key(0, root) self.split(temp, 0) self.insert_non_full(temp, k) else: self.insert_non_full(root, k) /* Insert non full condition */ def insert_non_full(self, x, k): i = len(x.keys) - 1 if x.leaf: x.keys.append((None, None)) while i >= 0 and k < x.keys[i]: x.keys[i + 1] = x.keys[i] i -= 1 x.keys[i + 1] = k else: while i >= 0 and k < x.keys[i]: i -= 1 i += 1 if len(x.child[i].keys) == (2 * self.t) - 1: self.split(x, i) if k > x.keys[i]: i += 1 self.insert_non_full(x.child[i], k) /* Split */ def split(self, x, i): t = self.t y = x.child[i] z = BTreeNode(y.leaf) x.child.insert_key(i + 1, z) x.keys.insert_key(i, y.keys[t - 1]) z.keys = y.keys[t: (2 * t) - 1] y.keys = y.keys[0: t - 1] if not y.leaf: z.child = y.child[t: 2 * t] y.child = y.child[0: t - 1] def main(): B = BTree(3) for i in range(10): B.insert_key((i, 2 * i)) B.print_tree(B.root) if B.search_key(8) is not None: print("nFound") else: print("nNot found") if __name__ == '__main__': main()
Searching Complexity on B Tree
Worst case Time complexity:
Average case Time complexity:
Best case Time complexity:
Average case Space complexity:
Worst case Space complexity:
B Tree Applications
- databases and file systems
- to store blocks of data (secondary storage media)
- multilevel indexing
Python Example for Beginners
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