Plants are able to keep growing indefinitely because they have tissues made of meristems—plant stem cells—which have the unique ability to transform themselves into the various specialized cells that make up the plant, dividing whenever appropriate and producing new cells of whatever type as needed. Meristems exist at the tips of all plants, allowing them to grow new stems or new roots, and, in trees, also in the trunk, where they add extra girth.
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Source: Phys.org