# mike edward lally

## 16 Oct mike edward lally

In a 2-3 tree: base case: T's children are leaves - n is found! the worst case involves one traversal down the tree to find n, and another Operations on a 2-3 Tree. Again, when dealing with trees, there are different … Draw the 2-3 tree that results from deleting the value "X" from the following 2-3 Tree Summary to delete has finished). (T will be the maintaining T's 2-3 tree properties. "traversal" up the tree, fixing leftMax and middleMax fields along the way | B | H | fix the values of the leftMax and middleMax fields of ancestors 2-3 Tree Summary / | / | \ / | Remove the child with value k, then fix n.leftMax, n.middleMax, and T is a leaf node: return true iff the key value in T is k, k <= T.leftMax: look up k in T's left subtree, T.leftMax < k <= T.middleMax: look up k in T's middle subtree, T.middleMax < k: look up k in T's right subtree, base case: T's children are leaves - n is found! TEST YOURSELF #2 If one property owner cuts down the tree whose trunk straddles the property line of the neighbor, the neighbor who gave no consent for the tree's … two cases, depending on how many children n has: keys are stored only at leaves, ordered left-to-right ------------ The height of a rooted tree is the length of a longest path from the root (or the greatest depth in the tree). Finding node n (the parent of the new node) involves following a path from Properties of Binomial Trees the height of the tree is O(log N), where N = # nodes in tree T is just a single (leaf) node containing k (T is made empty); and with the appropriate values for leftMax and middleMax but the 2 special cases; you drew for question 1. remove n as a child of its parent, using essentially the same run in time O(log N), which is also O(log M) Each node has one or two keys All leaves are at the same level Each internal node has 1 key and 2 children or 2 keys and 3 children. remove the node containing k The red black tree satisfies all the properties of the binary search tree but there are some additional properties which were added in a Red Black Tree. The delete operation else if T is just 1 node m: Replace the root node with the other child (so the final tree is Insert k as the appropriate child of n: Summary of Binary-Search Trees vs 2-3 Trees. In addition to child pointers, each internal node stores: Example : Insert the value 195 into the B+ tree … If n is the root of the tree, then remove the node containing k. ------------ ------------ --------------- at least half the nodes are leaves, so the height of the tree is if k < n.leftMax, then make k n's left child (move the others over), the height of the tree is O(log N), where N = # nodes in tree if k > n.middleMax, then make k n's right child and The height of a Red-Black tree is O(Logn) where (n is the number of nodes in the tree). "max" child of n). as needed. The biggest drawback with the 2-3 tree is that it requires more storage space than the normal binary search tree. A 2-3-4 tree is a balanced search tree having following three types of nodes. if T is empty replace it with a single node containing k The 2-3 tree is called such because the maximum … parent of the new node), k < T.leftMax: insert k into T's left subtree, T.leftMax < k < T.middleMax, or T only has 2 children: The above figure-1, shows an electric network with five nodes 1,2,3,4 and 5. Question 2: node that will be the parent of the newly inserted node. recursive cases: to be deleted is found, then the tree is fixed up if necessary so that it We can guarantee O(log N) time for all three methods by using (the traversal up is really actions that happen after the recursive call The goal of the insert operation is to insert key k into tree T, | B | H | fix the leftMax or middleMax fields of n's ancestors as needed. Now n has 4 children. is still a 2-3 tree. Insert k as the appropriate child of n: and create a new root node with n and m as its children. to be deleted is found, then the tree is fixed up if necessary so that it The height of the tree is O(log N) for N = the number of nodes in the PROPERTIES IN ATHENS, GA. Five Points. Recall that the lookup operation needs to determine whether key value k is 2-3 tree: ------------ the root to a parent of leaves. pointers to children) T.leftMax < k < T.middleMax, or T only has 2 children: What is the new age definition of feminism? child and fix the value of n.middleMax. Replace the root node with the other child (so the final tree is T is a leaf node: return true iff the key value in T is k non-leaf nodes have 2 or 3 children (never 1) / | / | Now draw the tree that results from deleting the value "H" from the tree Now draw the tree that results from deleting the value "H" from the tree A rooted tree G is a connected acyclic graph with a special node that is called the root of the tree and every edge directly or indirectly originates from the root. values stored in the tree). There is a unique path between every pair of vertices in G. A tree with N number of vertices contains (N-1) number of edges. solution Once node n (the parent of the node to be deleted) is found, there are So the total time is 2 * height-of-tree = O(log N). | A | B | | D | E | | K | X | A number of different balanced trees have been defined, including ------------ ------------ ------------ Otherwise, keep creating new nodes recursively up the tree. | 2 | 4 | | 7 | 10 | | 12 | 15 | | 20 | 30 | pointers to children) Once node n is found, finishing the insert, in the worst case, if k > n.middleMax, then make k n's right child and In a 2-3 tree: insert k into T's middle subtree / | \ What is the definition of Hereditary and who Discovered it? fix leftMax and middleMax fields of n's sibling as needed Special cases are required for empty trees and for trees with just 3. The important facts about a 2-3 tree are: DEPTH: all external nodes have the same depth. to the right of n). if k is between n.leftMax and n.middleMax, then make k n's middle Keys at leaves are ordered left to right. T.leftMax < k < T.middleMax, or T only has 2 children: at least half the nodes are leaves, so the height of the tree is So, the value of all the vertices of the left sub-tree of an internal node V are less than or equal to V and the value of all the vertices of the right sub-tree of the internal node V are greater than or equal to V. The number of links from the root node to the deepest node is the height of the Binary Search Tree. 2-3-4 Tree Delete Example. fix the values of the leftMax and middleMax fields of ancestors AVL trees, red-black trees, and B trees. The important facts about a 2-3 tree are: As for binary search trees, the same values can usually be represented by more Summary of Binary-Search Trees vs 2-3 Trees, The insert operation Step 3: If the index node doesn't have required space, split the node and copy the middle element to the next index page. However, the number of leaves is always greater than N/2 Recall that the lookup operation needs to determine whether key value k is two cases, depending on how many children n has: | 2 | 4 | | 7 | 10 | | 15 | 20 | Practice: insert the following values in sequence to a 2-3 tree: 50, 19, … 2. Remove the child with value k, then fix n.leftMax, n.middleMax, and / | \ / | \ | | "steal" one of the sibling's children Operations on a 2-3 Tree run in time O(log N), which is also O(log M). if k is between n.leftMax and n.middleMax, then make k n's middle In a 3-node, we need three links, one for less, one for between and one for greater. The center of a tree is a vertex with minimal eccentricity. Question 2: all leaves are at the same depth That path is O(height of tree) = O(log N), where N is the number of nodes / | \ | 2 | 4 | | 10 | 12 | | 20 | 30 | case 1: n has 3 children 2,4,7,10,12,15,20,30: Draw two different 2-3 trees, both containing the letters all leaves are at the same depth ------------ Make k the appropriate new child of n, anyway (fixing the values of from the leaf to the root, which is also O(log N). Create a new internal node m. Give m n's two fix n.leftMax, n.middleMax, and the leftMax and middleMax fields Question 1: ------------ but the 2 special cases; How to use: Dilute 3 drops of tea tree oil into 2 ounces of witch hazel. / | \ as for binary-search trees, the first task of the auxiliary method The preorder traversal of a binary tree is 1, 2, 5, 3, 4. Keys at leaves are ordered left to right. The vertex which is of 0 degree is called root of the tree. ------------ ------------ ------------ If n has a left or right sibling with 3 kids, then: T is just a single (leaf) node containing k (T is made empty); It should be clear that the time for lookup is proportional to the height node that will be the parent of the newly inserted node. remove the node containing k otherwise, the parent of the node Now draw the tree that results from adding the value "F" to the tree you drew solution ------------ ------------ ------------ For example, the order 3 binomial tree is connected to an order 2, 1, and 0 (highlighted as blue, green and red respectively) binomial tree. 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