Insert: B-Tree-Insert(T, k)

• Start at root(T) moving down the tree looking for the proper leaf to put k
• Split all full nodes along the way

B-Tree-Insert(T, k)
$\;\;\;\;\;$r = root(T)
$\;\;\;\;\;$if n(r) = 2t-1 $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$; full
$\;\;\;\;\;$then allocate empty node s pointing to r
$\;\;\;\;\;$ $\;\;\;\;\;$B-Tree-Split-Child(s, 1, r)
$\;\;\;\;\;$ $\;\;\;\;\;$B-Tree-Insert-Nonfull(s, k)
$\;\;\;\;\;$else B-Tree-Insert-Nonfull(r, k)

B-Tree-Insert-Nonfull(x, k)
$\;\;\;\;\;$if leaf(x)
$\;\;\;\;\;$then shift keys of x higher than k one to the right
$\;\;\;\;\;$ $\;\;\;\;\;$put k in appropriate spot
$\;\;\;\;\;$ $\;\;\;\;\;$n(x) = n(x) + 1
$\;\;\;\;\;$ $\;\;\;\;\;$DiskWrite(x)
$\;\;\;\;\;$else find smallest i such that k < key_i(x)
$\;\;\;\;\;$ $\;\;\;\;\;$DiskRead(child_i(x))
$\;\;\;\;\;$ $\;\;\;\;\;$if n(child_i(x)) = 2t - 1 ; full
$\;\;\;\;\;$ $\;\;\;\;\;$then B-Tree-Split-Child(x, i, child_i(x))
$\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$if k > key_i(x)
$\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$then i = i + 1 $\;\;\;\;\;$; adjust due to new node entry from child
$\;\;\;\;\;$ $\;\;\;\;\;$B-Tree-Insert-Nonfull(child_i(x), k)

Disk Accesses: O(h)

Run Time: O(th) = O(t $\log_t n$) = O(lg n), if t constant