There is no single botanical definition, which is part of what makes the question interesting.
What is a tree?
The working definition
Most definitions describe a tree as a woody plant with a single self-supporting trunk, a crown of branches and leaves, and an adult height of several metres. A common rule of thumb is a mature height of at least about 5 metres, but that is a convention and not a law.
Wood and growth rings
A tree's trunk thickens each year by adding a new layer of wood under the bark, beneath a growing layer called the cambium. In climates with seasons, the layers show as rings. Palms, bamboos and tree ferns look like trees but do not thicken in this way. Palms are monocots, like grasses.
Where it gets fuzzy
Hazels and many dogwoods are shrubs in some places and small trees in others. Some trees grow multi-stemmed when cut back or in poor conditions. Palms and tree ferns are called trees in everyday speech, though they are botanically quite different. Dendrologists accept that the line is practical rather than exact.
Two big groups
Most trees are either conifers, which carry seeds in cones and usually have needles or scales, or broadleaf trees, which are flowering plants and carry their seeds in fruits. In the timber trade the two are called softwoods and hardwoods, though that is a poor guide to how hard the wood actually is: balsa is a hardwood, and some yew is a hard softwood.
The parts of a tree
- Roots: anchor the tree, take up water and minerals and often connect to fungi that help the roots.
- Trunk: carries water up from the roots and sugars down from the leaves.
- Crown: branches and leaves, where light is captured.
- Bark: the protective outer layer.
- Flowers, fruit and cones: how the tree reproduces.
Why it matters for identification
Because trees differ so much in form, the best way to name one is to ignore its overall shape and work through its features in order. The key does that for you.