Tree (Johnny Duhan album)

Tree is an album by Irish folk singer Johnny Duhan.

Track listing

External links

2–3–4 tree

In computer science, a 2–3–4 tree (also called a 2–4 tree) is a self-balancing data structure that is commonly used to implement dictionaries. The numbers mean a tree where every node with children (internal node) has either two, three, or four child nodes:

2-node

2-node

3-node

3-node

4-node

4-node

2–3–4 trees are B-trees of order 4; like B-trees in general, they can search, insert and delete in O(log n) time. One property of a 2–3–4 tree is that all external nodes are at the same depth.

2–3–4 trees are an isometry of red–black trees, meaning that they are equivalent data structures. In other words, for every 2–3–4 tree, there exists at least one red–black tree with data elements in the same order. Moreover, insertion and deletion operations on 2–3–4 trees that cause node expansions, splits and merges are equivalent to the color-flipping and rotations in red–black trees. Introductions to red–black trees usually introduce 2–3–4 trees first, because they are conceptually simpler. 2–3–4 trees, however, can be difficult to implement in most programming languages because of the large number of special cases involved in operations on the tree. Red–black trees are simpler to implement, so tend to be used instead.

2–3 tree

In computer science, a 2–3 tree is a tree data structure, where every node with children (internal node) has either two children (2-node) and one data element or three children (3-nodes) and two data elements. Nodes on the outside of the tree (leaf nodes) have no children and one or two data elements. 2−3 trees were invented by John Hopcroft in 1970.

2 node

2 node

3 node

3 node

2–3 trees are an isometry of AA trees, meaning that they are equivalent data structures. In other words, for every 2–3 tree, there exists at least one AA tree with data elements in the same order. 2–3 trees are balanced, meaning that each right, center, and left subtree contains the same or close to the same amount of data.

Definitions

We say that an internal node is a 2-node if it has one data element and two children.

We say that an internal node is a 3-node if it has two data elements and three children.

We say that T is a 2-3 tree if and only if one of the following statements hold:

Mark

Mark may refer to:

Christianity

Currency

Mark (dinghy)

The Mark is a single-hander class of small sailing dinghy. The design probably first appeared in the 1960s, at about the same time as the Laser, but never took off as a popular racing class. The Mark is 12 feet (3.7 m) in length, with forward and side buoyancy compartments. A 19 feet (5.8 m) free standing rotating mast stepped far forward in the front buoyancy compartment supports a mainsail.

Gospel of Mark

The Gospel According to Mark (Greek: τὸ κατὰ Μᾶρκον εὐαγγέλιον, to kata Markon euangelion), the second book of the New Testament, is one of the four canonical gospels and the three synoptic gospels. It was traditionally thought to be an epitome (summary) of Matthew, which accounts for its place as the second gospel in the Bible, but most scholars now regard it as the earliest of the gospels. Most modern scholars reject the tradition which ascribes it to Mark the Evangelist, the companion of Peter, and regard it as the work of an unknown author working with various sources including collections of miracle stories, controversy stories, parables, and a passion narrative.

Mark tells of the ministry of Jesus from his baptism by John the Baptist to his death and burial and the discovery of the empty tomb – there is no genealogy or birth narrative, nor, in the original ending at chapter 16, any post-resurrection appearances. It portrays Jesus as a heroic man of action, an exorcist, healer and miracle worker. Jesus is also the Son of God, but he keeps his identity secret, concealing it in parables so that even the disciples fail to understand. All this is in keeping with prophecy, which foretold the fate of the messiah as Suffering Servant. The gospel ends, in its original version, with the discovery of the empty tomb, a promise to meet again in Galilee, and an unheeded instruction to spread the good news of the resurrection.