Graph vs. Tree — What's the Difference?

By Fiza Rafique & Urooj Arif — Updated on May 7, 2024
A graph is a general structure with nodes and edges connecting them, allowing loops and cycles. A tree is a hierarchical structure that's a specialized type of graph, with no cycles, always starting from a root node.

Key Differences

A graph is a collection of nodes connected by edges that can represent any kind of relationship. A tree, while also a graph, is a specific kind that represents hierarchical relationships.
A graph allows nodes to have multiple connections, including loops and cycles, whereas a tree has no cycles and every node has one parent except for the root.
In a graph, any node can be connected to any other node, whereas in a tree, there's always a unique path from the root to any other node.
Graphs can be either directed or undirected, meaning edges can represent one-way or two-way relationships, while trees are inherently directed, starting from a root and branching down.

Comparison Chart

Structure Type

Collection of nodes and edges
Hierarchical collection of nodes

Connectivity

Allows cycles and loops
No cycles, no loops

Directionality

Directed or undirected
Always directed

Root

No singular root node
Singular root node

Path

Multiple paths between nodes
Unique path from root to nodes

Compare with Definitions

Graph

A mathematical structure representing nodes connected by edges.
The social network graph reveals various user connections.

Tree

A graph often used to represent decision-making processes.
The decision tree helps classify outcomes efficiently.

Graph

A diagram showing data relationships visually.
The graph illustrates the increase in temperature over time.

Tree

A hierarchical graph with a root node and branching connections.
The organization's hierarchy resembles a tree structure.

Graph

A structure allowing for loops and multiple connections.
Complex networks often feature cycles within their graphs.

Tree

A graph with no cycles, ensuring a single path from root to nodes.
In a tree, each node has one unique parent.

Graph

A structure without a hierarchical root system.
Unlike a tree, a graph allows arbitrary connections.

Tree

A directed acyclic graph representing hierarchical relationships.
File systems often use tree structures to organize folders.

Graph

A structure that can be directed or undirected.
Road networks can be represented as directed graphs.

Tree

A data structure widely used in computer science.
Binary search trees speed up data retrieval operations.

Graph

A diagram that exhibits a relationship, often functional, between two sets of numbers as a set of points having coordinates determined by the relationship. Also called plot.

Tree

In botany, a tree is a perennial plant with an elongated stem, or trunk, supporting branches and leaves in most species. In some usages, the definition of a tree may be narrower, including only wood plants with secondary growth, plants that are usable as lumber or plants above a specified height.

Graph

A pictorial device, such as a pie chart or bar graph, used to illustrate quantitative relationships. Also called chart.

Tree

A woody perennial plant, typically having a single stem or trunk growing to a considerable height and bearing lateral branches at some distance from the ground.

Graph

The spelling of a word.

Tree

A wooden structure or part of a structure.

Graph

Any of the possible forms of a grapheme.

Tree

A thing that has a branching structure resembling that of a tree.

Graph

A written character that represents a vowel, consonant, syllable, word, or other expression and that cannot be further analyzed.

Tree

Force (a hunted animal) to take refuge in a tree.

Graph

To represent by a graph.

Tree

(of an area) planted with trees
Sparsely treed grasslands

Graph

To plot (a function) on a graph.

Tree

A perennial woody plant having a main trunk and usually a distinct crown.

Graph

A data chart (graphical representation of data) intended to illustrate the relationship between a set (or sets) of numbers (quantities, measurements or indicative numbers) and a reference set, whose elements are indexed to those of the former set(s) and may or may not be numbers.

Tree

An herbaceous plant or shrub resembling a tree in form or size.

Graph

(mathematics) A set of points constituting a graphical representation of a real function; (formally) a set of tuples $\left(x_1, x_2, \ldots, x_m, y\right)\in\R^\left\{m+1\right\}$, where $y=f\left(x_1, x_2, \ldots, x_m\right)$ for a given function $f: \R^m\rightarrow\R$. See also Graph of a function Category:en:Curves Category:en:Functions

Tree

Something that resembles a tree in form, especially a diagram or arrangement that has branches showing relationships of hierarchy or lineage.

Graph

(graph theory) A set of vertices (or nodes) connected together by edges; (formally) an ordered pair of sets $\left(V,E\right)$, where the elements of $V$ are called vertices or nodes and $E$ is a set of pairs (called edges) of elements of $V$. See also Graph (discrete mathematics)

Tree

(Computers) A structure for organizing or classifying data in which every item can be traced to a single origin through a unique path.

Graph

(topology) A topological space which represents some graph (ordered pair of sets) and which is constructed by representing the vertices as points and the edges as copies of the real interval [0,1] (where, for any given edge, 0 and 1 are identified with the points representing the two vertices) and equipping the result with a particular topology called the graph topology.

Tree

A wooden beam, post, stake, or bar used as part of a framework or structure.

Graph

A morphism $\Gamma_f$ from the domain of $f$ to the product of the domain and codomain of $f$, such that the first projection applied to $\Gamma_f$ equals the identity of the domain, and the second projection applied to $\Gamma_f$ is equal to $f$.

Graph

A graphical unit on the token-level, the abstracted fundamental shape of a character or letter as distinct from its ductus (realization in a particular typeface or handwriting on the instance-level) and as distinct by a grapheme on the type-level by not fundamentally distinguishing meaning.

A gallows.

Graph

(transitive) To draw a graph.

Tree

The cross on which Jesus was crucified.

Graph

To draw a graph of a function.

Tree

To force up a tree
Dogs treed the raccoon.

Graph

A curve or surface, the locus of a point whose coördinates are the variables in the equation of the locus; as, a graph of the exponential function.

Tree

(Informal) To force into a difficult position; corner
The reporters finally treed the mayor.

Graph

A diagram symbolizing a system of interrelations of variable quantities using points represented by spots, or by lines to represent the relations of continuous variables. More than one set of interrelations may be presented on one graph, in which case the spots or lines are typically distinguishable from each other, as by color, shape, thickness, continuity, etc. A diagram in which relationships between variables are represented by other visual means is sometimes called a graph, as in a bar graph, but may also be called a chart.

Tree

To supply or cover with trees
A hillside that is treed with oaks.

Graph

A drawing illustrating the relations between certain quantities plotted with reference to a set of axes

Tree

A perennial woody plant, not exactly defined, but differentiated from a shrub by its larger size (typically over a few meters in height) or growth habit, usually having a single (or few) main axis or trunk unbranched for some distance above the ground and a head of branches and foliage.
Hyperion is the tallest living tree in the world.
Birds have a nest in a tree in the garden.

Graph

Represent by means of a graph;
Chart the data

Tree

Any plant that is reminiscent of the above but not classified as a tree (in any botanical sense).
The banana tree

Graph

Plot upon a graph

Tree

An object made from a tree trunk and having multiple hooks or storage platforms.
He had the choice of buying a scratching post or a cat tree.

Tree

A device used to hold or stretch a shoe open.

Tree

The structural frame of a saddle.

Tree

(graph theory) A connected graph with no cycles or, if the graph is finite, equivalently a connected graph with n vertices and n−1 edges.

Tree

(computing theory) A recursive data structure in which each node has zero or more nodes as children.

Tree

(graphical user interface) A display or listing of entries or elements such that there are primary and secondary entries shown, usually linked by drawn lines or by indenting to the right.
We’ll show it as a tree list.

Tree

Any structure or construct having branches representing divergence or possible choices.
Family tree; skill tree

Tree

The structure or wooden frame used in the construction of a saddle used in horse riding.

Marijuana.

Tree

(obsolete) A cross or gallows.
Tyburn tree

Tree

(chemistry) A mass of crystals, aggregated in arborescent forms, obtained by precipitation of a metal from solution.

Tree

(cartomancy) The fifth Lenormand card.

Tree

(transitive) To chase (an animal or person) up a tree.
The dog treed the cat.

Tree

(transitive) To place in a tree.
Black bears can tree their cubs for protection, but grizzly bears cannot.

Tree

(transitive) To place upon a tree; to fit with a tree; to stretch upon a tree.
To tree a boot

Tree

(intransitive) To take refuge in a tree.

Tree

Any perennial woody plant of considerable size (usually over twenty feet high) and growing with a single trunk.

Tree

Something constructed in the form of, or considered as resembling, a tree, consisting of a stem, or stock, and branches; as, a genealogical tree.

Tree

A piece of timber, or something commonly made of timber; - used in composition, as in axletree, boottree, chesstree, crosstree, whiffletree, and the like.

Tree

A cross or gallows; as Tyburn tree.
[Jesus] whom they slew and hanged on a tree.

Tree

Wood; timber.
In a great house ben not only vessels of gold and of silver but also of tree and of earth.

Tree

A mass of crystals, aggregated in arborescent forms, obtained by precipitation of a metal from solution. See Lead tree, under Lead.

Tree

To drive to a tree; to cause to ascend a tree; as, a dog trees a squirrel.

Tree

A tall perennial woody plant having a main trunk and branches forming a distinct elevated crown; includes both gymnosperms and angiosperms

Tree

A figure that branches from a single root;
Genealogical tree

Tree

English actor and theatrical producer noted for his lavish productions of Shakespeare (1853-1917)

Tree

Chase a bear up a tree with dogs and kill it

Common Curiosities

Can a tree be considered a graph?

Yes, a tree is a special kind of graph without cycles and with a hierarchical structure.

Are all graphs connected?

Not necessarily; graphs can be disconnected, meaning there might not be paths between all nodes.

Can a tree have multiple roots?

No, a tree has a singular root node, unlike graphs which don't have a root.

Are trees always directed graphs?

Yes, trees are directed graphs, with all edges pointing away from the root node.

What types of trees are there in data structures?

Common types include binary trees, binary search trees, AVL trees, and red-black trees, each with specific properties and uses.

What is the purpose of using a graph?

Graphs are used to model relationships between entities, useful in various fields like social networking, transportation networks, and computer science.

Can a graph represent weighted relationships?

Yes, graphs can have weighted edges, where each edge has a value representing the strength or capacity of the connection.

How does a graph differ from a tree?

A graph allows arbitrary connections and cycles, while a tree has strict hierarchical rules with no cycles.

What is the difference between a rooted tree and a free tree?

A rooted tree has a designated root node that defines a hierarchical structure, whereas a free tree doesn't have a fixed root and can be viewed as undirected.

Are there limits to how many children a node in a tree can have?

In general trees, there are no specific limits. However, in binary trees, each node can have no more than two children.

What makes a graph cyclic or acyclic?

A graph is cyclic if it has at least one cycle, a path where the first and last nodes are the same. An acyclic graph has no such cycles.

How do directed and undirected graphs differ?

In directed graphs, edges have a direction indicating a one-way relationship, whereas undirected graphs have edges that represent two-way relationships.

How can a tree be traversed?

Trees can be traversed in various ways including in-order, pre-order, and post-order, each visiting nodes in a different sequence.

What is a spanning tree of a graph?

A spanning tree is a subset of a graph that includes all the vertices connected by the minimum number of edges needed to maintain connectivity without any cycles.

What is the depth of a tree?

The depth of a tree is the number of edges on the longest path from the root node to the furthest leaf node.

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