Grafting in the 19th century was a significant method of increasing stock of trees and plants as the photograph of East Anglian life by PH Emerson shows.  The objects of Grafting are to bring a bush or tree into an earlier fruiting than it would do naturally; to produce good fruit from an inferior plant; or to save space by putting dwarf scions on to rampant growing trees.
The stock or subject is the base tree and the scion or graft is the portion of the branch to be imported.

Gardeners Tips on Grafting

The best time to graft is in March when the sap is rising.
Suppress all buds that develop on the old stem below the graft.
Use a paper bag to offer some protection once the graft is done
The graft should be inclined slightly inward so the inner bark of the graft makes good contact with the stock.
Grafted vines should be packed in damp moss.
Fruit tree or rose shoots should be firm and well ripened. Grafting material should be from the previous years growth and slightly less advanced than the stock.

Methods of Grafting

Cleft, Wedge or Tongue-Grafting Cut off the top of the stock and cut out a thin wedge-shaped socket at the grafting point. Next cut the scion in a similar manner and wedge in and bind.

Double Grafting uses two scions but sacrifice the weaker one when they both take.

Theophrates Graft has all the branches cut from healthy root stock and tongues of a scions are slipped in to a slit, bound, and waxed.

Side Graft Make a T-shaped cut in the stem of the tree as deep as the inner bark.  Prepare the scion with a longitudinal sloping cut of the same length then insert and bind.

Approach grafting is used for obtaining choice varieties of vines, or   growing weaker vines on roots of a stronger grower.

Budding as practiced on roses only grafts a sliver of a bud to the stock plant but this can be done from May through August.

Other methods include Slit Graft, The Berlemboise Graft, Root Graft, Circle Graft, Rind Grafting Shield Grafting, or Herbaceous Graft for Pines, Oaks and Walnuts.

Give grafting a try, if you see a great specimen you can try grow a graft on a different root stock preferably from the same genus.