Henry V is quite rich in metaphors and analogies.

Henry alludes to the traditional Renaissance metaphor of the monarch as a sun. Just as the sun is the ruler of the heavens, so the king is the ruler of human society. This is what the King alludes to when he tells the French ambassadors that he will "rise there with so full a glory/ That I will dazzle all the eyes of France." (Act 1, scene 2, lines 278-79). He is like the rising sun in all its majesty. The Chorus echoes this when he describes how King Henry comforts and inspires his men on the  night before Agincourt. Like the sun, he provides warmth:

. . . every wretch, pining and pale before,  
Beholding him, plucks comfort from his looks.  
A largess universal, like the sun,  
His liberal eye doth give to everyone,  
Thawing cold fear . . .  
                                   Act 4, Chorus, lines 41-45

In Act 1, scene 2, the Archbishop of Canterbury makes a long analogy between a colony of honey-bees and a human society. Like humans, each bee colony has a king (in Shakespeare's time it was mistakenly thought that the queen bee was male), and the equivalent of magistrates, merchants, soldiers, masons, porters, and even executioners. The Archbishop's point is that in an ordered society everything has its proper function. Although the functions may be different, everything works to a common aim.

After the mocking gift of tennis balls from the Dauphin, King Henry replies with a series of puns that speak of warfare using the language of tennis:

When we have matched our rackets to these balls,  
We will in France, by God's grace, play a set  
Shall strike his father's crown into the hazard.  
Tell him he hath made a match with such a wrangler  
That all the courts of France will be disturbed  
With chases.  
                                     Act 1, scene 2, lines 261-66

The terms "crown," "hazard," and "chases" are all terms used in the royal tennis that was played in Shakespeare's time. They also have other meanings that can apply to the military situation. ("Chases" are military pursuits, for example.)

There are numerous more puns in the play. The bawdy humor in the scene between Catherine and her maid Alice, for example (Act 3 scene 4), depends on puns. And a careful reading of Burgundy's conversation with Henry in Act 5 scene 2, lines 272-304, shows that it contains a welter of puns and double-entendres.

In Act V, scene 2, lines 33-55, Burgundy constructs an extended metaphor of the state as a garden. In this case, the garden (France) has fallen into neglect (because of the ravages of war).