There are a number of ways the age of a plant or tree can be calculated, depending on whether the plant or tree is alive (and standing) or dead.
For dead trees/plants (fossils), carbon dating is the commonly used method, where the amount of carbon-14 remaining in the fossil gives an accurate estimate of its age. As mentioned, this method is used for fossils (things that have been dead for a while).
For a recently dead tree/plant, you can simply count the number of tree rings. Rings are developed at the rate of one per year, and a count of the rings will directly tell you its age.
For a tree/plant that is alive and standing, the age can be calculated without cutting it by using an increment borer. The borer bores into the tree up to its center and gives you an estimate of number of rings. Counting the rings will give the age in years.
I am not certain if you mean plant or planet?
For plants, determining the age is much easier for woody vegetation. The xylem and phloem tissue which is used to transport water and nutrients will generate "rings" growing from the center outward in the stem or trunk. The amount of rings gives a general age of the plant. Herbaceous plants are a little more complicated and sometimes will vary from species to species. However, a good general rule is to count the amount of limbs or observe plant height. If the plant has numerous limbs and is quite tall when compared to others, it can be said the plant is older.
If you are asking how the age of the planet is calculated, there are a few factors scientists will look into. First is fossil and element carbon dating, or half-life measurements. Elements take a while to degrade, and calculating a half-life allows us to understand how long a particular element has been on the planet. This is just a generalization to give us an estimate on the age of Earth.
I hope this helps out a little bit!