In this definition, a single vertex is counted as a path of length zero, and the same vertex may occur more than once within a path. It is reflexive: There is a trivial path of length zero from any vertex to itself. It is symmetric: If there is a path from u to v, the same edges form a path from v to u.

A trivial truth is simply a tautology of logic (sometimes with equality added). This means that a vacuous proof can only be possible in an inconsistent theory (because contradiction entails everything) while a trivial proof is a proof that is valid because formulation of the theorem is a tautology.

Since the zero solution is the “obvious” solution, hence it is called a trivial solution. Any solution which has at least one component non-zero (thereby making it a non-obvious solution) is termed as a “non-trivial” solution.

An acyclic graph is a graph having no graph cycles. Acyclic graphs are bipartite. A connected acyclic graph is known as a tree, and a possibly disconnected acyclic graph is known as a forest (i.e., a collection of trees). A graph with a single cycle is known as a unicyclic graph.

A polytree is a directed graph formed by orienting the edges of a free tree. Every polytree is a DAG.

A Tree is just a restricted form of a Graph. Trees have direction (parent / child relationships) and don’t contain cycles. They fit with in the category of Directed Acyclic Graphs (or a DAG). So Trees are DAGs with the restriction that a child can only have one parent.

Graph and tree are the non-linear data structure which is used to solve various complex problems. A graph is a group of vertices and edges where an edge connects a pair of vertices whereas a tree is considered as a minimally connected graph which must be connected and free from loops.

Check for a cycle with a simple depth-first search (starting from any vertex) – “If an unexplored edge leads to a node visited before, then the graph contains a cycle.” If there’s a cycle, it’s not a tree. If the above process leaves some vertices unexplored, it’s not a tree, because it’s not connected.

In graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph.

A complete representation without duplicate nodes is generated automatically. A procedure and software to implement the procedure have been devised to enable conversion from a tree representation to a graph representation of the requirements governing the development and design of an engineering system.

Like trees, graphs have nodes and edges. However, graphs are more general than trees: In a graph, a node can have any number of incoming edges (in a tree, the root node cannot have any incoming edges, and the other nodes can only have one incoming edge). Every tree is a graph, but not every graph is a tree.

Binary trees are used in Huffman coding, which are used as a compression code. Binary trees are used in Binary search trees, which are useful for maintaining records of data without much extra space.

A (mutable) binary tree, BiTree, can be in an empty state or a non-empty state: When it is empty, it contains no data. When it is not empty, it contains a data object called the root element, and 2 distinct BiTree objects called the left subtree and the right subtree.

A binary search tree (BST) is a node based binary tree data structure which has the following properties. The left subtree of a node contains only nodes with keys less than the node’s key. The right subtree of a node contains only nodes with keys greater than the node’s key.