Insertion into a B-tree
In this tutorial, you will learn how to insert a key into a btree. Also, you will find working examples of inserting keys into a B-tree in Python.
Inserting an element on a B-tree consists of two events: searching the appropriate node to insert the element and splitting the node if required.Insertion operation always takes place in the bottom-up approach.
Let us understand these events below.
- If the tree is empty, allocate a root node and insert the key.
- Update the allowed number of keys in the node.
- Search the appropriate node for insertion.
- If the node is full, follow the steps below.
- Insert the elements in increasing order.
- Now, there are elements greater than its limit. So, split at the median.
- Push the median key upwards and make the left keys as a left child and the right keys as a right child.
- If the node is not full, follow the steps below.
- Insert the node in increasing order.
Let us understand the insertion operation with the illustrations below.
The elements to be inserted are 8, 9, 10, 11, 15, 16, 17, 18, 20, 23.
Algorithm for Inserting an Element
BreeInsertion(T, k) r root[T] if n[r] = 2t - 1 s = AllocateNode() root[T] = s leaf[s] = FALSE n[s] <- 0 c1[s] <- r BtreeSplitChild(s, 1, r) BtreeInsertNonFull(s, k) else BtreeInsertNonFull(r, k) BtreeInsertNonFull(x, k) i = n[x] if leaf[x] while i ≥ 1 and k < keyi[x] keyi+1 [x] = keyi[x] i = i - 1 keyi+1[x] = k n[x] = n[x] + 1 else while i ≥ 1 and k < keyi[x] i = i - 1 i = i + 1 if n[ci[x]] == 2t - 1 BtreeSplitChild(x, i, ci[x]) if k &rt; keyi[x] i = i + 1 BtreeInsertNonFull(ci[x], k) BtreeSplitChild(x, i) BtreeSplitChild(x, i, y) z = AllocateNode() leaf[z] = leaf[y] n[z] = t - 1 for j = 1 to t - 1 keyj[z] = keyj+t[y] if not leaf [y] for j = 1 to t cj[z] = cj + t[y] n[y] = t - 1 for j = n[x] + 1 to i + 1 cj+1[x] = cj[x] ci+1[x] = z for j = n[x] to i keyj+1[x] = keyj[x] keyi[x] = keyt[y] n[x] = n[x] + 1
/* Inserting a key on a B-tree in Python */ /* Create a node */ class BTreeNode: def __init__(self, leaf=False): self.leaf = leaf self.keys =  self.child =  /* Tree */ class BTree: def __init__(self, t): self.root = BTreeNode(True) self.t = t /* Insert node */ def insert(self, k): root = self.root if len(root.keys) == (2 * self.t) - 1: temp = BTreeNode() self.root = temp temp.child.insert(0, root) self.split_child(temp, 0) self.insert_non_full(temp, k) else: self.insert_non_full(root, k) /* Insert nonfull */ 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_child(x, i) if k > x.keys[i]: i += 1 self.insert_non_full(x.child[i], k) /* Split the child */ def split_child(self, x, i): t = self.t y = x.child[i] z = BTreeNode(y.leaf) x.child.insert(i + 1, z) x.keys.insert(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] /* 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) def main(): B = BTree(3) for i in range(10): B.insert((i, 2 * i)) B.print_tree(B.root) if __name__ == '__main__': main()
Python Example for Beginners
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