Hello Lisa,

What a beautiful cherry tree!  I can understand why you would want to try to propagate it for your granddaughter. 

There are several methods to use to propagate with a tree branch, and the choice is yours to make. It can be a  difficult process, but often successful using a rooting hormone (available in garden centers in powder or gel form), using one year old branches, maintaining high humidity/moisture and inserting the 1/3 of cutting below the surface.  

The techniques I read about propagating generally suggest this to be accomplished in the early summer to early autumn when the branch is firm and green, selecting and removing a branch about the thickness of a pencil (4-6 inches long is  recommended). Remove the leaves from the bottom half, leaving a few on the top. Scrape the bark off the bottom inch of the stem to expose the green (cambium) layer creating a "wound".  Place the wounded end into a rooting hormone. Plant the cutting 2 inches deep in well drained medium such as perlite, sand or a mixture of peat moss and potting soil. You can cover the pot with a plastic bottle to create a greenhouse effect, although not necessary, keeping the new cutting it in bright, indirect sunlight is essential. Roots take 4-12 weeks, sometimes longer. 

Air layering does seem to offer the best results, and highest success rate. The success is deemed more likely as it forces the branch to grow roots while still attached to the parent tree. And it appears to be easier to manage and see the progress of the roots. First, girdle the stem, remove several leaves around the wound, pack with sphagnum or peat moss, cover with polyethylene plastic and tie each end. Keep the moss moist until roots form. Rooting hormone can be applied to the wounded area. 

We do not make a recommendation for you, just supply the information for you to decide. However, in all the reading I did for this  question,  air layering seemed the most effective for success.

The links below are informative on propagation methods and techniques.  

https://extension.unh.edu/sites/default/files/migrated_unmanaged_files/Resource002093_Rep3093.pdf

https://www.purdue.edu/hla/sites/yardandgarden/extpub/new-plants-from-cuttings-text-only/

https://pubs.extension.wsu.edu/product/propagating-deciduous-and-evergreen-shrubs-trees-vines-with-stem-cuttings/

https://yardandgarden.extension.iastate.edu/how-to/propagation-trees-shrubs-and-vines-hardwood-cuttings

I wish you luck in this endeavor. I hope this information is  helpful and useful. Please reach out if you have any further questions. All the best in your gardening and recreating the tree for your granddaughter. .