"After being planted/transplanted for some time, the tree has established."
So how is tree establishment defined? The tree could be described as having growth in height or of new shoots, or increase in girth, or both. Some have included root growth, while others simply observed whether trees have survived.
There are three practical points in determining that trees have established or not, by observing shoot growth, trunk movement in rootball and dieback (or lack of it) in the crown. These three criteria can be applied practically and do not require specialised equipment.
So how is root growth and spread, or trunk movement in rootball gauged?
Trunks that show movement at ground level may suggest poor root establishment, which usually leads to poor stability. The Burnley Method (Leers et al. 2018, Moore 1985) is a field test that involves force applied forward and back by a human onto the trunk, in order to determine root establishment and tree stability. The human would hold onto the trunk at a height that is above half of the tree's height and the trunk is moved/flexed 0.6 to 1.0m forward and back, and with not so much force as to snap the trunk. If the trunk is observed to be moving at or below soil surface, it is an indication of a pooly established root system.
Leers, M., Moore G.M., and May P.M. 2018. Assessment of six indicators of street tree establisment in Melbourne, Australia. Arboriculture & Urban Forestry 44:12-22.
Moore, G.M. 1985. Getting to the roots of the problem. The International Plant Propagators Society, Combined Proceedings 35:105-111.